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v V Amel’kin [ ZDifferential Equationsin Applications\"

MirPublishersMoscow

Translated from the Russian by Eugene YankovskyFirst published 1990Revised from the 1987 Russian editionHa anenuiicxom xsanePrinted in the Union of Soviet Socialist RepublicsISBN 5-03-000521-8© I/Iairaremsorno <<Hay1<a». Fnannaa penarmna|i}H3HI{0·M£lT€M&TH'I€CKOi nnreparypu, 1987© English translation, Eugene Yankovsky, 1990

ContentsPreface 7Chapter 1. Construction of Differen-tial Models and Their Solutions 111.1 Whose Coffee Was Hotter? 111.2 Steady-State Heat Flow . . . 141.3 An Incident in a National Park 181.4 Liquid Flow Out of Vessels. TheWater Clock . . . . 261.5 Effectiveness of Advertising 301.6 Supply and Demand 321.7 Chemical Reactions . . . 341.8 Differential Models in Ecology 381.9 A Problem from the MathematicalTheory of Epidemics 441.10 The Pursuit Curve 511.11 Combat Models . . . 551.12 Why Are Pendulum Clocks Inaccu-rate? ....... 691.13 The Cycloidal Clock . . . 731.14 The Brachistochrone Problem 811.15 The Arithmetic Mean, the Geo-metric Mean, and the AssociatedDifferential Equation .... 881.16 On the Flight of an Object Thrownat an Ang e to the Horizon 931.17 Weightlessness ......... 961.18 Kep er’s Laws of Planetary Motion 1001.19 Beam Deflection . . . 1121.20 Transportation of Logs 119Chapter 2. Qualitative Methods ofStudying Differential Models . . 1362.1 Curves of Constant Direction ofMagnetic Needle ........ 136

6 Contents2.2 Why Must an Engineer KnowExistence and Uniqueness Theo-rems? . . . . . . . 1432.3 A Dynamical Interpretation ofSecond-Order Differential Equa-tions...’ . ... 1552.4 Conservative Systems in Mechanics 1622.5 Stability of Equilibrium Pointsand of Periodic Motion 1762.6 Lyapunov Functions . . . 1842.7 Simple States of Equilibrium . 1892.8 Motion of a Unit—Mass ObjectUnder the Action of LinearSprings in a Medium with LinearDrag . . . . . . . 1952.9 Adiabatic Flow of a Perfect GasThrough a Nozzle of VaryingCross Section . . ..... 2032.10 Higher-Order Points of Equilib-rium . . . . . . . 2112.11 Inversion with Respect to a Circleand Homogeneous Coordinates 2182.12 Flow of a Perfect Gas Through aRotating Tube of Uniform CrossSection . ..... 2232.13 Isolated Closed Trajectories . 2362.14 Periodic Modes in Electric Circuits 2502.15 Curves Without Contact . . 2602.16 The Topographical System ofCurves. The Contact Curve . . . 2632.17 The Divorgence of a Vector Fieldand Limit Cycles 270Selected Readings 274Appendices 275Appendix 1. Derivatives of Elemen-tary Functions . . 275Appendix 2. Basic Integrals .,.. . 278

PrefaceDifferential equations belong to one of themain mathematical concepts. They areequations for finding functions whose deriv-atives (or differentials) satisfy"given condi-tions. The differential equations arrived atin the process of studying a real phenomenonor process are called the differential modelof this phenomenon or process. It is clearthat differential models constitute a particu-lar case of the numerous mathematical modelsthat can be built as a result of studies ofthe world that surrounds us. It must beemphasized that there are different types ofdifferential models. This book considers nonebut models described by what is knownas ordinary differential equations, one char-acteristic of which is that the unknownfunctions in these equations depend ona single variable.In constructing ordinary differential mo-dels it is important to know the laws ofthe branch of science relating to the natureof the problem being studied. For instance,in mechanics these may be Newton’s laws,in the theory of electric circuits Kirchhoff’slaws, in the theory of chemical reactionrates the law of mass action.Of course, in practical life we often haveto deal with cases where the flaws that en-

8 Prefaceable building a differentia lequation (or sev-eral differential equations) are not known,and we must resort to various assumptions(hypotheses) concerning the course of theprocess at small variations of the parame-ters, the variables. Passage to the limitwill then lead to a differential equation,and if it so happens that the results ofinvestigation of the differential equation asthe mathematical model agree with the ex-perimental data, this will mean that thehypothesis underlying the model reflectsthe true situation.*When working on this book, I had twogoals in mind. The first was to use examples,rich in content rather than purely illustra-tive, taken from various fields of knowledgeso as to demonstrate the possibilities ofusing ordinary differential equations ingaining an understanding of the world aboutus. Of course, the examples far from ex-haust the scope of problems solvable by or-dinary differential equations. They givean idea of the role that ordinary differentialequations play in solving practical problems.The second goal was to acquaint the read-* If the reader wishes to know more about mathe-matical models, he can turn to the fascinating booksby A.N. Tikhonov and D.P. Kostomarov, StoriesAbout Applied Mathematics (Moscow: Nauka,1979) (in Russian), and N .N. Moiseev, Mathemat-igs Stages an Experiment (Moscow: Nauka, 1979)(nn Russian).

Preface 9er with the simplest tools and methodsused in studying ordinary differential equa-tions and characteristic of the qualitativetheory of diyjerential equations. The fact isthat only in rare cases are we able to solvea differential equation in the so-calledclosed form, that is, represent the solutionas a formula that employs a finite numberof the simplest operations involving ele-mentary functions, even when it is knownthat the differential equation has a solu-tion. In other words, we can say that thegreat variety of solutions to differentialequations is such that for their representa-tion in closed form a finite number of ana-lytical operations is insufficient. A simi-lar situation exists in the theory of algebraicequations: while for first- and second-orderalgebraic equations the solutions can alwaysbe easily expressed in terms of radicalsand for third- and fourth-order equationsthe solutions can still be expressed in termsof radicals (although the formulas becomevery complicated), for a general algebraicequation of an order higher than the fourththe solution cannot be expressed in radicals.To return to differential equations. Ifan infinite series of this or that form is usedto represent the solutions, then the scopeof solvable equations broadens considerab-ly. Unfortunately, it often happens thatthe most essential and interesting proper-ties of the solutions cannot be revealed by

10 Prefacestudying the form of the series. More thanthat, even if a differential equation can besolved in closed form, more often than notit is impossible to analyze such a solutionsince the relationship between the variousparameters of the solution often provesto be extremely complicated.This shows how important it is to developmethods that make it possible to acquirethe data on the various properties of thesolutions without solving the differentialequations themselves. And indeed, suchmethods do exist. They constitute the es-sence of the qualitative theory of differen-tial equations, which is based on the gener-al theorems regarding the existence anduniqueness of solutions and the continuousdependence of solutions on the initial dataand parameters. The role of existence anduniqueness theorems is partially discussedin Section 2.2. As for the general qualitativetheory of ordinary differential equations,ever since J.H. Poincaré and A.M. Lya-punov laid the foundations at the end ofthe 19th century, the theory has been inten-sively developing and its methods are wide-ly used when studying the world about us.I am indebted to Professors Yu.S. Bog-danov and M.V. Fedoryuk for the construc-tive remarks and comments expressed inthe process of preparing the book for publi-cation.. . l _ V.V, Amefkin

Chapter 1Construct1onof D1fferent1al Modelsand Their Solut1ons1.1 Whose Coffee Was Hotter?When Tom and Dick ordered coffee andcream in a lunch room, they were given bothsimultaneously and proceeded as follows.Tom poured some of his cream into the cof-fee, covered the cup with a paper napkin,and went to make a phone call. Dick cov-ered his cup with a napkin and poured thesame amount of cream into his coffee onlyafter 10 minutes, when Tom returned. Thetwo started drinking their coffee at tl1e sametime. Whose was hotter?We will solve this problem on the naturalassumption that according to the laws ofphysics heat transfer through the surface ofthe table and the paper napkin is much lessthan through the sides of the cups and thatthe temperature of the vapor above the sur-face of the coffee in the cups equals the tem-perature of the coffee.We start by deriving a relationship indi-cating the time dependence of the tempera-ture of the coffee in Dick’s cup before thecream is added.In accordance with our assumption onthe basis of a law of physics, the amount

12 Differential Equations in Applicationsof heat transferred to the air from Dick’scup is determined by the formuladQ Z qui}?-Sat, (1)where T is the coffee’s temperature at timet, 6 the temperature of the air in the lunchroom, 1] the thermal conductivity of thematerial of the cup, l the thickness of thecup, and s the area of the cup’s lateral sur-face. The amount of heat given off by thecoffee isdQ = —-cm dT, (2)where c is the specific heat capacity of thecoffee, and m the mass (or amount) of coffeein the cup. If we now consider Eqs. (1)and (2) together, we arrive at the followingequation:q J-`{it-dr: -cmdr,which, after variable separation, can be re-written as follows:dT __ qs———-T_6 — —T:;—- dt. (3)Denoting the initial temperature of the coffeeby T0 and integrating the differential equa-tion (3), we find thatT=6—|—(T0—9)exp(—T;l’%—t), (4)This formula is the analytical expressionof the law whereby the temperature of the

Ch. 1. Construction of Differential Models 13coffee in Dick’s cup varied prior to the addi-tion of the cream.[Now let us establish the law whereby thetemperature of the coffee in Dick’s cupchanged after Dick poured in cream. Weuse the heat balance equation, which herecan be written ascm (T — Bo) = ¢1mi(6D — Ti). (5)where BD is the temperature of the Dick’scoffee with cream at time t, T1 the temper-ature of the cream, C1 the specific heat ca-pacity of the cream, and ml the mass ofcream added to the coffee.Equation (5) yieldsi Clml Cm9D— cm-|—c1m1 Tl-l_ cm—|—c1m1 T` (6)Bearing in mind formula (4), we can re-write (6) as follows:HD CTn+C1m1 1 + cm+c1m1- ....I|f.>< [a+<T.. ¤>¤xp( ,,,,, ¢)]. <7>which constitutes the law whereby the tem-perature of the coffee in Dick’s cup variesafter cream is added.To derive the law for temperature varia-tion of the coffee in Tom’s cup we againemploy the heat balance equation, whichnow assumes the formCm (T0 " 90) = cimi (90 "" T1): (8)

14 Differential Equations in Applicationswhere 00 is the temperature of the mixture.lf we solve (8) for 00, we getc m em69: cmil-dlml 11*+ cm—|—c1m1 TO'Then, using Eq. (4) with 0,, serving as theinitial temperature and cm + clml substi-tuted for cm., we arrive at the law for thetemperature (0T) variation of the coffeein Tom’s cup as analytically given by thefollowing formula:"_ Clml - Cm L9T`"`9+li ¢m—|—c1m1 Ti T ¢m—|—c1m1 To 9].. _.._..._..*lSX €Xp( l(¢m+ Cimi) tl ° (9)Thus, to answer the question posed inthe problem we need only turn to formulas(7) and (9) and carry out the necessary cal-culations, bearing in mind that C1 z 3.9 ><103 J/kg·K, cxé.1 ><103 J/kg·K, andnz0.6 V/m·K and assuming, for the sakeof definiteness, that ml : 2 >< 10*2 kg,m=8 >< 10*2 kg, T1 =20°C, 0 :20°C,T0 =80°C, s = 11 ><10*3 mz, and Z:2 X 10*3 m. The calculations show thatTom’s coffee was hotter.1.2 Steady-State Heat FlowThe reader will recall that a steady-stateheat flow is- one in which the object’s tem-

Ch. 1. Construction of Differential Models 15‘ l \\§ w\ u \% n \§ it N§ · axn 2 isni n in20Fig. 1 Fig. 2perature at each point does not vary withtime.II'1 pI'Ol'Jl€I1'lS Wll0SB physical COI`ll,€Ilt· isrelated to the effects of heat flows, an impor-tant role is played by the so—called isother-mal surfaces. To clarify this statement, letUS COI1Sld€I’ 3 h€3l»··COIldllCtlI1g (Flg`U.I'€1) 20 cm in diameter, made of a homogene-ous material, and protected by a layer ofmagnesium oxide 10 cm thick. We assumethat the temperature of the pipe is 160 °Cand the outer surface of the protective cov-ering has a temperature of 30 °C. It isintuitively clear then that there is a surface,designated by a dashed curve in Figure2, at each point of which the temperatureis the same, say 95 °C. The dashed curve inFigure 2 is known as an isotherm, while the

16 Differential Equations in Applicationssurface corresponding to this curveiis knownas an isothermal surface. In general, iso-therms may have various shapes, depend-ing, for one, on the nonsteady-state natureof the heatiflow and on the nonhomogeneityof the material. In the case at hand theisotherms (isothermal surfaces) are represent-ed by concentric circles (cylinders).We wish to derive the law of temperaturedistribution inside the protective coatingand find the amount of heat released by thepipe over a section 1 m long in the courseof 24 hours, assuming that the thermal con-ductivity coefficient lc is equal to 1.7 ><10*4.To this end we turn to the Fourier law,according to which the amount of heat re-leased per unit time by an object that is in sta-ble thermal state and whose temperature T ateach point is solely a function of coordinate xcan be found according to the formuladTQ = -—kF (ac) -8-5:- == const, (IO)where F (x) is the cross-sectional area normalto the direction of heat flow, and lc the thermalconductivity coefficient.The statement of the problem impliesthat F (:2:) = 2;r•;xl, where l is the length ofthe pipe (cm), and sc the radius of the baseof the cylindrical surface lying inside theouter cylinder. Then on the basis of (IO)

Ch. 1. Construction of Differential Models 17we gets0 Q 20 dIS dT: _ 0.00017><2nz S T’ (M)160 10T Q x dgS dt: " 0.00017><2nz S (12)160 10Integrating (11) and (12) yields160-T _ ln 0.1x __ log 0.1.1:130 " 1n2 "' Iog2 °HenceT = 591.8 —· 431.8 log .7:.This formula expresses the law of tempera-ture distribution inside the protectivecoating. We see that the length of the pipeplays no role in this law.To answer the second question, we turnto Eq. (11). Then for l = 100 cm we findthatQ ___ 130><0.00017><2 ><100— 1n2_ 200n><130><0.00017" 0.69315 ’and, hence, the amount of heat released bythe pipe in the course of 24 hours is 24 X60 X GOQ -·= 726 852 J.2-0770

18 Differential Equations in Applications1.3 An Incident in a National ParkWhile patrolling a national park,two for-est rangers found the carcass- of a slain wildboar. Inspection showed that the poacher’sshot had been precise and the boar had diedon the spot. Reasoning that the poachershould return for the kill, the rangers de-cided to wait for him and hid nearby. Soontwo men appeared, heading directly for thedead boar. When confronted by the rangers,the men denied having anything to do withthe poaching. But by this time the rangershad collected indirect evidence of theirguilt. It only remained to establish theexact time of the kill. This they did byusing the law of heat emission. Let us seewhat reasoning this involved.According to the law of heat emission,the rate at which an object cools off in airis proportional to the difference betweenthe temperature of the object and that ofthe air,2%.: -k (x-G), (13)where x is the temperature of the object attime t, a the temperature of the air, andk a positive proportionality coefficient.Solution of the problem lies in an analy-sis of the relationship that results from in-tegrating the differential equation (13).Here one must bear in mind that after the

Ch. 1. Construction of Differential Models 19boar was killed, the temperature of the aircould have remained constant but alsocould have varied with time. In the first caseintegration of the differential equation (13)with variables separable leads to1n%-5%: —kt, xqéa, (14)where xo is the temperature of the objectat time t = O. Then, if at the time whenthe strangers were confronted by the rangersthe temperature of the carcass, x, was 31 °Cand after an hour 29 °C and if when the shotwas fired the temperature of the boar was:1; = 37 °C and the temperature of the aira = 21 °C, we can establish when the shotwas fired by putting t = O as the time whenthe strangers were detained. Using thesedata and Eq. (14), we find that31--21Now, substituting this value of lc and .2: =37 into (14), we gett— 14.; In" 0.22314 31-211- ·—--OE-—@ ln 1.6: -2.10630.In other words, roughly 2 hours and 6 min-utes passed between the time the boar waskilled and the time the strangers were de-tained.2*

20 Diierential Equations in ApplicationsIn the second case, that is, when the tem-perature of the air varies in time, the cool-ing off of the carcass is expressed by thefollowing nonhomogeneous linear differen-tial equation$35- —}— kx: lca(t), (16)where 0, (t) is the temperature of the air attime 13.To illustrate one of the methods for deter-mining the time when the boar was killed,let us assume that the temperature of thecarcass was 30 °C when the strangers weredetained. Let us also suppose that it isknown that on the day of the kill the tem-perature of the air dropped by 1 °C everyhour in the afternoon and was 0 °C whenthe carcass was discovered. We will alsoassume that after an hour had passed afterthe discovery the temperature of the carcasswas 25 °C and that of the air was downto ——1° C. If we now assume that the shotwas fired at t=0 and that :1:0 = 37 °Cat t = 0, we get a (t) = t* —— t, wheret-= 15* is the time when the rangers dis-covered the carcass.Integrating Eq. (16), we getx=(37—t—lc·1)e·"’—|—t* —- t —{- k’1.If we 11ow bear in mind that sc = 30 °Cat t=t* and :c=25°C at t-==t*-{-1,

Ch. 1. Construction of Differential Models 21the last formula yields(37 —- t* ——- Irl) exp (——-Ict*) -l- Irl == 30,(37 — t* - Irl) exp [—k (t* -}- 1)]""‘ kl-1 7-' 26.These two equations can be used to derivean equation for Ic, namely,(30 -— Irl)e·’* — 26 + Irl = 0. (17)We can arrive at the same equation start-ing from different assumptions. Indeed,suppose that at if: 0 the carcass of the slainboar was found. Then a (t) == ——t andwe arrive at the differential equation13%+ kx: -kz (18)(with the initial data 2:,, = 30 at t= 0),from which we must {ind as as an explicitfunction of t.Solving Eq. (18), we getx = (30 —— Irl) exp (——Ict) -—— t + Irl. (19)Setting sc = 25 and t = 1 in this relation-ship, we again arrive at Eq. (17), which ena-bles solving the initial problem numerical-ly.Indeed, as is known, Eq. (17) cannot besolved algebraically for Ic. But it is easilysolved by numerical methods for finding theroots of transcendental equations, for one,

22 Differential Equations in Applicationsby Newton’s method of approximation. Thismethod, as well as other methods of succes-sive approximations, is a way of using arough estimate of the true value of a rootused to obtain more exact values of theroot. The process can be continued untilthe required accuracy is achieved.To show how Newton’s method is used,we transform Eq. (17) to the form30k — 1 —|- (1 — 26k) exp (lc) = 0, (20)and Eq. (19), setting :1: = 37, to the form(37k — 1 —l- kt) exp (kt) — 30k —l— 1 = 0.(21)Both equations, (20) and (21), are of thetype(asc —|— b) exp (kx) -i— cx —{— cl = 0. (22)If we denote the left-hand side of Eq. (22)by qa (sc), differentiation with respect to .2:yieldscp' (sc) = (lax —[— Ptb -|- a) exp (M:) + c,cp" (x) = (Wax —{- Vb -1- 2M) exp (kx).Then according to Newton’s method forfinding a root of Eq. (22), if for the ithapproximation xi we have the inequality(P ($2) (P" (xi) > 0»tht? D€XlL &pPI`0XlIIl&l&l0U, -76;+1, C3I1 130 f0U.Hd

Ch. 1. Construction of Differential Models 23via the formulax•xi+1 = xi •To directly calculate the root (to within,say, one part in a million), we compilethe following program using BASIC*:10 CLS:PRINT "Solution of equation by New-ton’s method"15 INPUT "lambda=", L;20 INPUT "a==", A: INPUT "b=", B;30 INPUT "c=", C: INPUT "d=", D40 INPUT "approximate value of root-.=", X50 PRINT "X", "f", "f"’, "f""100 E = EXP (L*X)110 F = (A*X -1- B)*E + C*X + D120 F1 = (L*(A*X -1- B) -1- A)*E -1- C130 F2 = L* (L* (A*X -1- B) -1- 2*A)*E150 PRINT X, F, F1, F2·i51 IF F = 0 THEN END155 IF F1 = 0 THEN PRINT "Newton’s methodis divergent": END170 IF F*F2<0 THEN PRINT "Newton’smethod is divergent": END190 X = X —- F/F1200 GOTO 100In this program X, f, f' and f" stand forkn, cp (kn), cp' (kn), and cp" (kn), respective-ly.* Edit0r’s note. Some BASIC lines, e.g. line10, have been split due to the printed format;they should be input as one line.

24 Differential Equations in ApplicationsAfter starting the program enter the re-quested values of the coefficients of the equa-tion and the initial value of the root. Theresults can be listed in a table of the approx-imate values of the root and the respectivevalues of the function and its iirst andsecond derivatives._ Employing this general procedure, let usturn to Eq. (20).Difierentiating its left-handside cp (k) with respect to lc, we arrive atcp' (lc) = 30 - (25 -|— 26}.:) exp (k).It can the11 easily be verified that tp (0) =0, cp (1) < 0, and q>’ (0) > 0. Thus, thefunction cp increases in a small neighborhoodof the origin and then dec1·eases to a nega-tive value at lc = 1. This implies that in theinterval (0, 1) there is a root of the equationcp (k) : 0. To hnd this root we run the pro-gram. Below we give the protocol of theprogram:Solution of equation by Newton’s methodlambda: 1a=—26 b=1c=30 d=—1Approximate value of root = 0.5X f f' f".5 -5.784655 -32.65141 -105.5182.322836 -1.525956 -16.11805 -82.02506.2281622 -.3514252 -8.859806 -71.52333.1884971 -5.519438E-02 -6.103379 -67.49665.1794539 ·—~2.748013E·—-03 -5.497021 ——-66.60768

Ch. 1. Construction of Differential Models 25.178954 -7.629395E-06 -5.46373 -66.55884.1789526 -4.768372E-07 -5.463642 -66.5587.1789525 0 -5.463635 -66.5587OKIn this protocol X, f, f', and f” stand forkn, rp (kn), cp' (kn), and cp" (kn), respective-ly.The final step in solving the problem con-sists in substituting the calculated valuekn z k z 0.178952 into Eq. (21) and solv-ing the latter for t (the time when the wildboar was killed). To employ the abovescheme, we denote the left—hand side of Eq.(21) by g (t). Then, selecting -1 for thevalue of to and bearing in mind that inthis case a =k,b : 37k-1,c =0,d=-30k-l—1, and ft:-k, we can find the timewhen the boar was killed. To this endwe find the coefficients b and d, which proveto be equal to 5.621243 and -4.368575, re-spectively, and then running the above pro-gram, find the sought time t. The proto-col of the program is given below:Solution of equation by Newton’s methodlambda = 0.1789525a =-·-· 0.1789525 b = 5.621243 c = 0d = -4.368575Approximate value of root = -1X f f' f"-1 .181972 .9639622 .1992802-·-1.188775 3.506184E-03 .9270549 .1917861

26 Differential Equations in Applications`csHFig. 3-1.192557 1.430512E-06 .9263298 .1916388-1.192559 0 .9263295 .1916387OKIn this protocol X, f, f', and f" stand fortn, g (tn), g' (tn), and g" (tn), respectively.These results imply that the boar waskilled approximately l hour and 12 minutesbefore the rangers discovered the carcass.1.4 Liquid Flow Out of Vessels.The Water ClockThe two problems that we now discuss illu-strate the relationship between the physicalcontent of a problem and geometry. But firstlet us examine some general theoretical con-clusions.We take a vessel (Figure 3) whose hori-zontal cross section has an area that is afunction of the distance from the bottomof the vessel to the cross Section. Suppose

Ch. 1. Construction of Differential Models 27that initially at time t = O the level of theliquid in the vessel is at a height of hmeters. We will also suppose that the areaof the vessel’s cross section at height xis denoted by S (:2:) and that the area of theopening in the bottom of the vessel is s.As is known, the rate v at which the liquidflows out of the vessel at the moment when theliquid’s level is aiheight .2: is given by the for-mula v == kl/2g:c, where g = 9.8 m/sz,and k is the rate constant of the outflow pro-cess.In the course of an infinitesimal timeinterval dt the outflow of the liquid can beassumed uniform, whereby during dt acolumn of liquid with a height of v dtand a cross-sectional area of s will flow outof the vessel, which causes the level of theliquid to change by -——dx (the "minus" be-cause the level lowers).The above reasoning leads us to the fol-lowing differential equationks dt == —S (:1:) dx,which can be rewritten asat T.- ...`-LY-$*9..;.-. d . 23ks \/2gx at ( )Let us now solve the following problem.A cylindrical vessel with a vertical axissix meters high and four meters in diameterhas a circular opening in the bottom. The

26 Differential Equations in Applications`csHFig. 3-1.192557 1.430512E—06 .9263298 .1916388-4.192559 0 .9263295 .1916387OKIn this protocol X, f, f', and f" stand fortn, g (tn), g' (tn), and g" (tn), respectively.These results imply that the boar waskilled approximately 1 hour and 12 minutesbefore the rangers discovered the carcass.1.4 Liquid Flow Out of Vessels.The Water ClockThe two problems that we now discuss illu-strate the relationship between the physicalcontent of a problem and geometry. But firstlet us examine some general theoretical con-clusions.We take a vessel (Figure 3) whose hori-zontal cross section has an area that is afunction of the distance from the bottomof the vessel to the cross section. Suppose

Ch. 1. Construction of Differential Models 27that initially at time t = 0 the level of theliquid in the vessel is at a height of hmeters. We will also suppose that the areaof the vessel’s cross section at height agis denoted by S (ac) and that the area of theopening in the bottom of the vessel is s.As is known, the rate v at which the liquidflows out of the vessel at the moment when theliquid’s level is aiheight x is given by the for-mula v = kl/2gx, where g = 9.8 m/sz,and k is the rate constant of the outflow pro-cess.In the course of an infinitesimal timeinterval dt the outflow of the liquid can beassumed uniform, whereby during dt acolumn of liquid with a height of v diand a cross—sectional area of s will flow outof the vessel, which causes the level of theliquid to change by —d:c (the “minus" be-cause the level lowers).The above reasoning leads us to the fol-lowing difierential equationmi/QE at Z -s ix) ax,which can be rewritten asdz : ___...*H"L. d .ks 1/2g:c x (23)Let us now solve the following problem,A cylindrical vessel with a vertical axissix meters high and four meters in diameterhas a circular opening in the bottom. The

28 Differential Equations in Applicationsradius of this opening is 1/12 m. Find howthe level of water in the vessel depends ontime t and the time it takes all the water toflow out. TBy hypothesis, S (.1:) = 4at and s =1/144. Since for water k = 0.6, Eq. (23)assumes the form_ __ 217.152 dxdt V;Integrating this differential equation yieldst= 434.304 [ l/6 — l/EL Ogxgfi,which is the sought dependence of the levelof water in the vessel on time t. If we putx = 0 in the last formula, we find that ittakes approximately 18 minutes for all thewater to flow out of the vessel.Now a second problem. An ancient waterclock consists of a bowl with a small holein the bottom through which water flowsout of the bowl (Figure 4). Such clocks wereused in ancient Greek and Roman courts totime the lawyers’ speeches, so as to avoidprolonged speeches. We wish to determinethe shape of the water clock that would en-sure that the water level lowered at a con-stant rate.This problem can easily be solved viaEq. (23). We rewrite this equation in theform· _ __ S (=¤> Q;

Ch. 1. Construction of Differential Models 29.z·§

‘ YFig. 4Precisely, if we suppose that the bowl hasthe shape of a surface of revolution, in ac-cordance with the notations used in Figure4, Eq. (24) yields the following result:1/ 2% Z —--2 a, <25>ks ]/2gwhere a = vx ·—-= da:/dt is the projection onthe :1: axis of the rate of motion of thewater’s free surface, which is constant byhypothesis. Squaring both sides of Eq.(25), we arrive at the equationx = cr", (26)with c = aznz/2gk2s”. The latter means thatthe sought shape of the water clock is ob-tained by rotating curve (26) about the 2:axis.

30 Differential Equations in Applications1.5 Effectiveness of AdvertisingSuppose that a retail chain is selling com-modities of a certain type, say B, aboutwhich only rz: buyers out of N potential buy-ers know at time t. Let us also assume thatto speed up sales the chain has placed pro-motion materials in the local TV and radionetwork. All further information about thecommodities is distributed among the buy-ers via personal contact between them. Wemay assume with a high probability thatafter the TV and radio network have re-leased the information about B, the rate ofchange in the number of persons knowingabout B is proportional both to the numberof buyers knowing about B and to the num-ber of buyers not knowing about B.If we suppose that time is reckoned fromthe moment when the promotion materialsand advertisements were released andN /·y persons have learned about the commo-dities, we arrive at the following differen-tial equation:%:kx (N- 2:), (27)with the initial condition that sc = N /yat t = O. In this equation lc is a positiveproportionality factor. Integrating, we findthat‘%·‘ ln + C.

Ch. 1. Construction of Differential Models 31scxv0 tFig. 5Assuming that NC =· C1, we arrive at theequation;..... NmN_x -Ae ,with A = eC¤. When solved for sr, thisequation yieldsNm"’=N —%.·r‘·‘=_Lv‘w·r · (28)Ae +1 1+1>a· *with P = Cl/A.In economics, Eq. (28) is commonlyknown as the logistic-curve equation.If we allow for the initial condition, Eq.(28) becomesN35 = ——————-—-— .1+<v—4>e·"'*‘Figure 5 provides a schematic ofa logis-tic curve for y = 2. In conclusion we note

32 Differential Equations in Applicationsthat the problem of dissemination of tech-nological innovations can also be reducedto Eq. (27).1.6 Supply and DemandAs is known, supply and demand constituteeconomic categories in a commodity econo-my, categories that emerge and function onthe market, that is, in the sphere of trade.The first category represents the commodi-ties that exist on the market or can be deliv-ered to it, while the second represents thedemand for commodities on the market.One of the main laws of a market economyis the law of supply and demand, whichcan be formulated simply by saying thatfor each commodity some price must existthat will cause the supply and the demandto be just equal. Such a price establishes an"equilibrium" on the market.Let us consider the following problem.Suppose that in the course of a (relativelylong) time interval a farmer sells his produce(say, apples) on the market, and that hedoes this immediately after the apple har-vest has been taken in and then once eachfollowing week (i.e. with a week’s interval).The farmer’s stock of apples being fixed af-ter he has collected his harvest, the week’ssupply will depend on both the expectedprice of apples in the coming week and theexpected change in price in the following

Ch. 1. Construction of Differential Models 33weeks. If it is assumed that in the comingweek the price of apples will fall and in thefollowing weeks it will grow, then supplywill be restrained provided that the ex-pected rise in price exceeds storage costs. Inthese conditions the greater the expected risein price in the following weeks, the lowerthe supply of apples on the market. On theother hand, if the price of apples in thecoming week is high and then a fall is ex-pected in the following weeks, then thegreater the expected fall in price in the fol-lowing weeks, the higher the supply of ap-ples on the market in the coming week.If we denote the price of apples in thecoming week by p and the time derivativeof price (the tendency of price formation)by p', both supply and demand are func-tions of these quantities. As practice hasshown, depending on various factors thesupply and demand of a commodity may berepresented by different functions of priceand tendency of price formation. For exam-ple, one function is given by a linear de-pendence expressed mathematically in theform y = ap' + bp —l- c, where a, b, andc are real numbers (constants). Say, inour example, the price of apples in thecoming week is taken at 1 rouble for 1 kg,in t weeks it was p (t) roubles for 1 kg,and the supply s and demand q are givenby the functionss=44p’ —}—2p—1, q==4p’-—2p—|—39.3-0770

34 Differential Equations in ApplicationsThen, for the supply and demand to be inequilibrium, we must require that44p’ —l-2p—1=4p' —2p—§—39.This condition leads us to the differentialequationdp __Integration yields p = Cermt -}— 10. If weallow for the initial condition that p = 1at t = 0, the equilibrium price is given bythe formulap = —9e·1°’ + 10. (29)Thus, if we want the supply and demand tobe in equilibrium all the time, the pricemust vary according to formula (29).1.7 Chemical ReactionsA chemical equation shows how the inter-action of substances produces a new sub-stance. Take, for instance, the equation2H2 —{— O2 -—>2H2O.It shows how as a result of the interactionof two molecules of hydrogen and one mo-lecule of oxygen two molecules of water areformed.

Ch. 1. Construction of Differential Models 35Generally a chemical reaction can be writ-ten in the formaA —i- bB —l- cC —l- ——>mM + nN+ pP +where A, B, C, are molecules of the in-teracting substances, M, N, P, . . mole-cules of the reaction products, and the con-stants a, b, c, . ., m, n, p, . positiveintegers that stand for the number of mole-cules participating in the reaction.The rate at which a new substance isformed in a reaction is called the reactionrate, and the active mass or concentrationof a reacting substance is given by the num-ber of moles of this substance per unit vol-ume.One of the basic laws of the theory ofchemical reaction rates is the law of massaction, according to which the rate of a chem-ical reaction proceeding at a constant tem-perature is proportional to the product of theconcentrations of the substances taking partin the reaction at a given moment.Let us solve the following problem. Twoliquid chemical substances A and B occu-pying a volume of 10 and 20 liters, respec-tively, form as a result of a chemical reactiona new liquid chemical substance C. Assum-ing that the temperature does not changein the process of the reaction and that everytwo volumes of substance A and one vol-3* r

36 Differential Equations in Applicationsume of substanceBform three volumes ofsubstance C, {ind the amount of substanceC at an arbitrary moment of time t if ittakes 20 min to form 6 liters of C.Let us denote by x the volume (in liters)of substance C that has formed by the timet(in hours). By hypothesis, by that time2.1:/3 liters of substance A and x/3 liters ofsubstance B will have reacted. This meansthat we are left with 10 -— 2x/3 liters of Aand 20 ·—— x/3 liters of B. Thus, in accord-ance with the law of mass action, we arriveat the following differential equationdx 2::: rcw:K(‘°"T) (Z0-?)which can be rewritten asdxwhere k is the proportionality factor (lc =2K/9). We must also bear in mind that sinceinitially (t = 0) there was no substanceC, we may assume that x = 0 at t = 0. Asfor the moment t = 1/3, we have sc = 6.Thus, the solution of the initial problemhas been reduced to the solution of a so-called boundary value problem:ft;-:k (15-x) (60-:c), x (0): 0,

Ch. 1. Construction of Differential Models 37To solve it, we iirst integrate the differen-tial equation and allow for the initial con-dition x(0) :0. As a result we arriveat the relationship_?_g_;_;’_ Z 4€45m_Since :1: = 6 at t = 1/3, substituting thesevalues into the relationship yields emh =3/2. Hence,_T4(€15k)3t:;4(3/2)3t,that is,$:15 i-(2/3)***i·—(i/@(2/3)3’ lThis gives the amount of substance Cthat has been formed in the reaction bytime t.A remark is in order. From practicalconsiderations it is clear that only a finitevolume of substance C can be formed when10 liters of A interact chemically with 20liters of B. However, a formal study ofthe above relationship between x and tshows that for a finite t, namely at (2/3);** =4, the variable .1: becomes iniinite. But thisfact does not contradict practical consid-erations because it is realized only for anegative value of t, while the chemical reac-tion is considered only for t nonnegative.

38 Differential Equations in Applications1.8 Differential Models in EcologyEcology studies the interaction of man and,in general, living organisms with the envi-ronment. The basic object in ecology is theevolution of populations. Below we describedifferential models of populations that dealwith their reproduction or extinction andwith the coexistence of various species ofanimals in the predator vs. prey situation.*Let x (t) be the number of individualsin a population at time t. If A is the numberof individuals in the population that areborn per unit time and B the number ofindividuals that die off per unit time, thenthere is reason to say that the rate at whichx varies in time is given by the formuladx—(F..A—B. (30)The problem consists in finding the depen-dence of A and B on :1:. The simplest situa-tion is the one in whichA = ax, B = bx, (31)where a and b are the coefficients of birthsand deaths of individuals per unit time,respectively. Allowing for (31), we can re-write the differential equation (30) asdxTi-T -. (cz-- b) as. (32)* For more detail see J.D. Murray, "Some simplemathematical models in ecology," Math. Spectrum16, No. 2: 48-54 (1983/84).

Ch. 1. Construction of Differential Models 30Assuming that at t = to the number of in-dividuals in the population is sc = :1:,,and solving Eq. (32), we get:1: (t) = .2:0 exp [(a —— b) (t — t,,)l.This implies that if a > b, then the numberof individuals sc tends to infinity as t—> oo,but if a<b, then a:——>O as t-+00 andthe population becomes extinct.Although the above model is a simplifiedone, it still reflects the actual situation insome cases. However, practically all mo-dels describing real phenomena and proces-ses are nonlinear, and instead of the differ-ential equation (32) we are forced to con-sider an equation of the typedx __jg--f (iv),where f (.2:) is a nonlinear function, say theequation%—·:f(x) =ax-—bx2,wherea > 0, b > O.Assuming that x = xo at t = to and solvingthe last equation, we get; : _ 33“’ ll ¤=0+<¤/b—x0>exp1—¢»<¢—¢0>1 ( )We see that as t —> oo the number of indiv-iduals in the population, x (t), tends to

40 Differential Equations in ApplicationsczJ? ·5>.1201-zz--1***-@""”'”*°—0 b <.Z‘0 6Fig. 6a/b. Two cases are possible here: a/b > xoand a/b < xo. The difference between thetwo is clearly seen in Figure 6. Note thatformula (33) describes, for one thing, thepopulations of fruit pests and some typesof bacteria.If we consider several coexisting species,say, big and small {ish, where the small fishserve as prey for the big, then by setting updifferential equations for each species wearrive at a system of equations%?—=f, (1:,, x,,), i=1, 2, n.Let us study in greater detail the two-spe-cies predator vs. prey model, first intro-duced by the Italian mathematician VitoVolterra (1860-1940) to explain the oscilla-tions in catch volumes in the Adriatic Sea,which have the same period but differ inphase.

Ch. 1. Construction of Differential Models 41Let ac be the number of big fish (predators)that feed on small fish (prey), whose num-ber we denote by y. Then the number ofpredator fish will grow as long as they havesufficient food, that is, prey fish. Finally, asituation will emerge in which there willnot be enough food and the number of bigHsh will diminish. As a result, starting froma certain moment the number of small {ishbegins to increase. This, in turn, assists anew growth in the number of the big spe-cies, and the cycle is repeated. The modelconstructed by Volterra has the formda:Y: —a:c—|—b:cy, (34)dy -— asTH-.-cx—dxy, ( )where ez, b, c, and d are positive constants.In Eq. (34) for the big iish the term bxyreflects the dependence of the increase inthe number of big fish on the number ofsmall {ish, while in Eq. (35) the term——d;z:y expresses the decrease in the numberof small fish as a function of the number ofbig {ish.To make the study of these two equationsmore convenient we introduce dimension-less variables:d bu(t)=-Fx, v(·•:)=—&—y, 1:=ct, ot=a/c.

42 Differential Equations in ApplicationsAs a result the differential equations (34)and (35) assume the formu,' : ow. (v — 1), v' : v (1 -- u), (36)with oc positive and the prime standing fordifferentiation with respect to 1.Let us assume that at time rc = to thenumber of individuals of both species isknown, that is,u (T0) = wo, v (10) = va (37)Note that we are interested only in positivesolutions. Let us establish the relation-ship between u and v. To this end we dividethe first equation in system (36) by the sec-ond and integrate the resulting differentialequation. We getow+u—-lnv°°u =ow0—i-un-—lnvg°u0EH,where H is a constant determined by theinitial conditions (37) and parameter oc.Figure 7 depicts the dependence of uon v for different values of H. We see thatthe (u, v)—plane contains only closed curves.Let us now assume that the initial val-ues uo and vt, are specified by point Aon the trajectory that corresponds to thevalue H = H3. Since un > 1 and vo < 1,the first equation in (36) shows that at firstvariable u, decreases. The same is true ofvariable v. Then, as u approaches a valueclose to unity, v' vanishes, which is fol-lowed by a prolonged period of time 1: in the

Ch. 1. Construction of Differential Models 43uA. H5Hz’ {D"'1I0 1 vFig. 7u vv00 Mdo0 z·Fig. 8course of which variable v increases. Whenv becomes equal to unity, uf vanishes andvariable u begins to increase. Thus, bothu and v traverse a closed trajectory. Thismeans that the solutions are functions peri-odic in time. The variable u does not achieve

44 Differential Equations in Applicationsits maximum when v does, that is, oscil-lations in the populations occur in differ-ent phases. A typical graph of u and v asfunctions of time is shown in Figure 8 (forthe case where v0>1 and u0<1).In conclusion we note that the study ofcommunities that interact in a more com-plex manner provides results that are moreinteresting from the practical standpointthan those obtained above. For example, iftwo populations fight for the same source offood (the third population), it can be demon-strated that one of them will becomeextinct. Clearly, if this is the third popula-tion (the source of food), then the other twowill also become extinct.1.9 A Problem from theMathematical Theory of EpidemicsLet us consider a differential model encoun-tered in the theory of epidemics. Supposethat a population consisting of N individ-uals is split into three groups. The first in-clude individuals that are susceptible to acertain disease but are healthy. We denotethe number of such individuals at timet by S (t). The second group incorporatesindividuals that are infected, that is, theyare ill and serve as a source of infection.We denote the number of such idividualsin the population at time t by I (t). Final-ly, the third group consists of individuals

Ch. 1. Construction of Differential Models 45who are healthy and immune to the di-sease. We denote the number of such indivi-duals at time t by R (t). Thus,S (t) —l— I (t) —l— R (t) = N. (38)Let us also assume that when the numberof infected individuals exceeds a certainfixed number I *, the rate of change in thenumber of individuals susceptible to thedisease is proportional to the number ofsuch individuals. As for the rate of changein the number of infected individuals thateventually recover, we assume that it isproportional to the number of infected in-dividuals. Clearly, these assumptions sim-plify matters and in a number of cases theyreflect the real situation. Because of thefirst assumption, we suppose that when thenumber of infected individuals I (t) isgreater than I *, they can infect the individ-uals susceptible to the disease. This meansthat we have allowed for the fact that theinfected individuals have been isolated fora certain time interval (as a result of quar-antine or because they have been far fromindividuals susceptible to the disease).We, therefore, arrive at the following dif-ferential equation-—ocS if I t > I*,%:l 0 `fI()·" * (39)I § I .Now, since each individual susceptibleto the disease eventually falls ill and be-

46 Differential Equations in Applicationscomes a source of infection, the rate ofchange in the number of infected individualsis the difference, per unit time, between thenewly infected individuals and those thatare getting well. Hence,dj {ocS——B1 if 1(t) > 1*, 40dt --BI if 1(t) { 1*. ( )We will call the proportionality factorsot and B the coefyicients of illness and recovery.Finally, the rate of change of the numberof recovering individuals is given by theequation dR/dt -·= BI.For the solutions of the respective equa-tions to be unique, we must hx the initialconditions. For the sake of simplicity weassume that at time t = O no individualsin the population are immune to the disease,that is, R (O) = O, and that initially thenumber of infected individuals was I (0).Next we assume that the illness and recov-ery coefficients are equal, that is, on = B(the reader is advised to study the casewhere ot =;& B). Hence, we find ourselves ina situation in which we must consider twocases.Case 1. 1 (O)< 1*. With the passage oftime the individuals in the population willnot become infected because in this casedS/dt = O, and, hence, in accordance withEq. (38) and the condition R (O) == O,we have an equation valid for all values

Ch. 1. Construction of Differential Models 47N . ----——--—--—————-———- ·-S(é)If—-·-···········"‘""""" *‘"' "“""'/?(¢)IO5)0 itFig. 9of tzS(t)==S(O)=N——I(0).The case considered here corresponds to thesituation when a fairly large number of in-fected individuals are placed in quaran-tine. Equation. (40) then leads us to the fol-lowing differential equationdI7]-;--- ——otI.This means that I (t) = I (0) e‘°°* and,hence,R(t)=N—S(t)——I(t)= I (O) [1 — e‘°°*].Figure 9 provides diagrams that illustratethe changes with the passage of time in the

48 Differential Equations in Applicationsnumber of individuals in each of the threegroups.Case 2. I (0) > I*. In this case there mustexist a time interval OQ t < T in whichI (t) > I* for all values of t, since by itsvery meaning the function I = I (t) iscontinuous. This implies that for all t’sbelonging to the interval [O, T] the diseasespreads to the individuals susceptible to it.Equation (39), therefore, implies thatS (t) = S (0)e‘°°'for OQ t < T. Substituting this into Eq.(40), we arrive at the differential equationij;--}-0tI =otS (O) e·°°’. (41)If we now multiply both sides of Eq. (41)by em, we get the equationé. (lem) = as (0).Hence, Ie°°’ = otS (0) t —i- C and, there-fore, the set of all solutions to Eq. (41) isgiven by the formulaI (t) = Ce"°°* —l— cxS (0) te‘°°’. (42)Assuming that t = O, we get C = I (0) and,hence, Eq. (42) takes the formI (t) = [I (0) + 0tS (O) tl e'°", (43)with Og t < T.

Ch. 1. Construction of Differential Models 49We devote our further investigations tothe problem of finding the specific value ofT and the moment of time tmax at which thenumber of infected individuals provesmaximal.The answer to the first question is impor-tant because starting from T the individualssusceptible to the disease cease to becomeinfected. If we turn to Eq. (43), the afore-said implies that at t= T its right-handside assumes the value 1*, that is,I* = [I (O) -1- ocS (O) Tle‘°°T. (44)ButS (T) = 1imS (t) =S (oo)f->00is the number of individuals that are suscep-tible to the disease and yet avoid fallingill; for such individuals the following chainof equalities holds true:S (T) = S (oo) = S (0) e‘°°T.Hence,__ 1 S (0)Thus, if we can point to a definite valueof S (oo), we can use Eq. (45) to predict thetime when the epidemic will stop. Substitut-ing T given by (45) into Eq. (44), we arriveat the equation* = S (0) S (°°)I [I (O)—|-S(O) ln Sho) il Sw) ,4-0770

50 Differential Equations in Applicationsor.i’L.. = i& ifiS(0¤) $(0) +1H S(00) ’which can be rewritten in the form1* __ I (0)——S(cO) —§—1nS(0o) --—————S(0) -}-1nS (0). (46)Since I * and all the terms on the right-handside of (46) are known, we can use this equa-tion to determine S (oo).To answer the second question, we turnto Eq. (43). In accordance with the posedquestion, this equation yieldsig-=(as (0) - a1 (0) »- azs (0) q e—<¤ :0.The moment in time at which I attains itsmaximal value is given by the following for-mula:... _i_ _ I (0)tmu “‘ ¤= li S0)If we now substitute this value into Eq. (43),we getIm..x= S(0)¤X1>{—li -—-’ (0)/S (0>l}= S (tmax)•This relationship shows, for one, that attime tmax the number of individuals sus-ceptible to the disease coincides with thenumber of infected individuals.

Ch. 1. Construction of Differential Models 51N ·······**""""*‘***“""' **** """"“"""““’“‘’' " ''`"""'“`““`'' —R(t}[ __-ml _ mv- — smI0 Ht)tH7U.Z' T ZFig. 10But if t > T, the individuals susceptibleto the disease cease to be infected andI (t) =I*e*°¤(*·T).Figure 10 gives a rough sketch of the dia-grams that reflect the changes with the pas-sage of time of the number of individualsin each of the three groups considered.1.10 The Pursuit CurveLet us examine an example in which differ-ential equations are used to choose a cor·rect strategy in pursuit problems.Qi

52 Differential Equat.ions in Applicationsd8 »/I xfi \\¤ \\\xdrr if0Fig. 11We assume that a destroyer is in pursuitof a submarine in a dense fog. At a certainmoment in time the fog lifts and the sub-marine is spotted floating on the surfacethree miles from the destroyer. The destroy-er’s speed is twice the submarine’s speed.We wish to {ind the trajectory (the pursuitcurve) that the destroyer must follow topass exactly over the submarine if the latterimmediately dives after being detectedand proceeds at top speed and in a straightline in an unknown direction.To solve the problem we first introducepolar coordinates, r and 9, in such a mannerthat the pole O is located at the point atwhich the submarine was located when dis-covered and that the point at which thedestroyer was located when the submarinewas discovered lies on the polar axis r(Figure 11). Further reasoning is based on

Ch. 1. Construction of Differential Models 53the following considerations. First, the de-stroyer must take up a position in which itwill be at the same distance from pole Oas the submarine. Then it must move aboutO in such a way that both moving objectsremain at the same distance from point O allthe time. Only in this case will the destroyereventually pass over the submarine whencircling pole O. The aforesaid implies thatthe destroyer must first go straight to pointO until it finds itself at the same distancex from O as the submarine.Obviously, distance x can be found eitherfrom the equation27 _ 3--:1:72-- 2vor from the equationx _ 3-]-::T `"` -5- ’where v is the submarine’s speed a11d 20the destr0yer’s speed. Solving these equa-tions, we find that this distance is either onemile or three miles.Now, if the two still have not met, thedestroyer must circle pole O (clockwise orcounterclockwise) and head away from thepole at a speed equal to that of the subma-rine, v. Let us decompose the destroyer’s ve-locity vector (of length 2v) into two compo-nents, the radial component U,. and the tan-gential component vt (Figure 11).

54 Diflerential Equations in ApplicationsThe radial component is the speed atwhich the destroyer moves away from poleO, that is,, - 2;I. _· ’while the tangential component is the li-near speed at which the destroyer circlesthe pole. The latter component, as is known,is equal to the product of the angular ve—locity dt)/dt and radius r, that is,d0"t=*a;·But since v, must be equal to v, we havevt:]/(2v)2-02:]/-3:0.Thus, the solution of the initial problemis reduced to the solution of a system of twodifferential equations,dr d9 ··-52- ..v, r W - V 3 v,which, in turn, can be reduced to a singleequation, dr/r = d9/ V3, by excluding thevariable t.Solving the last differential equation, wefind thatr=Ce°/Vg,with C an arbitrary constant. o

Ch. 1. Construction of Differential Models 55If we now allow for the fact that destroyerbeginsits motion about pole O starting fromthe polar axis r at a distance of sz: miles awayfromO, thatis,r = 1 at 0 = 0 and r = 3 at9 =—:n;, we conclude that in the first caseii C = 1 and in the second C = 3e”/V; Thus,to fulfill its mission, the destroyer mustmove two or six miles along a straight linetoward the point where the submarine wasdiscovered and then move in the spiralr = e9/VT or the spiral r = 3e(9+”)/V?1.11 Combat ModelsDuring the First World War the Britishengineer and mathematician FrederickW. Lanchester (1868-1946) constructed sev-eral mathematical models of air battles.Later these models were generalized so as todescribe battles involving regular troops orguerilla forces or the two simultaneously.Below we consider these three models.Suppose that two opposing forces, nvand y, are in combat. We denote the num-ber of personnel at time t measured in daysstarting from the first day of combat opera-tions by sz: (t) and y (t), respectively. It isthe number of personnel that will play adecisive part in the construction of thesemodels, since it is difficult, practicallyspeaking, to specify the criteria that wouldallow taking into account, when comparingthe opposing sides, not only the number of

56 Differential Equations in Applicationspersonnel but also combat readiness, levelof military equipment, preparedness andexperience of the officers, morale, and agreat many other factors.We also assume that x(t) and y(t) changecontinuously and, more than that, aredifferentiable as functions of time. Of course,these assumptions simplify the real sit-uation because as (t) and y (t) are integers.But at the same time it is clear that if thenumber of personnel on each side is great,an increase by one or two persons will havean infinitesimal effect from the practicalstandpoint if compared to the effect pro-duced by the entire force. Therefore, we mayassume that during small time intervalsthe change in the number of personnel isalso small (and does not constitute an inte-gral number). The above reasoning is, ofcourse, insufficient to specify concrete for-mulas for ;z: (t) and y (t) as functions of t, butwe can already point to a number of factorsthat enable describing the rate of changein the number of personnel on the twosides. Precisely, we denote by OLR the rateat which side x suffers losses from diseaseand other factors not related directly tocombat operations, and by CLR the rateat which it suffers losses directly in combatoperations involving side y. Finally by RRwe denote the rate at which reinforcementsare supplied to side x. It is then clear thatthe total rate of change of x (t) is given by

Ch. 1. Construction of Differential Models 57the equation.<l=é1£l = - (OLR +01,12) + RR. (47)A similar equation holds true for y (t).The problem now is to {ind the appropriateformulas for the quantities OLR, CLR,and RR and then examine the resultingdifferential equations. The results shouldindicate the probable winner.We use the following notation: a, b,c, d, g, and h are nonnegative constantscharacterizing the effect of various factorson personnel losses on both sides (x and y),P (t) and O (t) are terms that allow for thepossibility of reinforcements being suppliedto :2: and y in the course of one day, andxo and yo are the personnel in :1: and y priorto combat. We are now ready to set up thethree combat models suggested by Lan-chester.* The iirst refers to combat opera-tions involving regular troops:rw: —a$ (t)""b!/ U) +P (t)»9%-gl:-cx(:)-dy(¢)+Q(z).* Examples of combat operations are given in thearticle by C.S. Coleman, "Combat models" (seeDijjercntial Equation Models, eds. M. Braun,C.S. Coleman, and D.A. Drew, New York: Sprin-ger, 1983, pp. 109-131). The article shows howthe examples agree with the models considered.

58 Differential Equations in ApplicationsIn what follows we call this system the A-type differential system (or simply the A-type system).The second model, specified by the equa-tionsd<r(¢> ..q;··——··¢$(¢)—€¢¤(¢)y(¢)+P(¢)·d i!jg%—=—dy(*)—h¢(¢)y(¢)+Q(¢)describes combat operations involving onlyguerilla forces. We will call this system theB-type system. Finally, the third model,which we call the C-type system, has theformd··%%Q= ··¤¤*¤ (t)-—.€’¤¤(¢)y(¢)+P(¢)»d—%€L= -· cw (¢)—dy(¢> +Q (¢)and describes combat operations involvingboth regular troops and guerilla forces.Each of these differential equationsdescribes the rate of change in the numberof personnel on the opposing sides as a func-tion of various factors and has the form(47). Losses in personnel that are not direct-ly related to combat operations and are de-termined by the terms —a.2: (t) and—dy (t) make it possible to describe theconstant fractional loss rates (in the absenceof combat operations and reinforcements)

Ch. 1. Construction of Differential Models 59via the equations1 dx ___ — 1 Q_ _YHT" a’ Y dz "' d'If the Lanchester models contain onlyterms corresponding to reinforcements andlosses not associated with combat opera-tions, this means that there are no com-bat operations, while the presence of theterms —by (t), —c;z: (t), -gx (t) y (t), and—-hx (t) y (t) means that combat operationstake place.In considering the A-type system, weassume, first, that each side is within therange of the fire weapons of the other and,second, that only the personnel directlyinvolved in combat come under fire. Undersuch assumptions Lanchester introduced theterm -——by (t) for the regular troops of sidezz: to reflect combat losses. The factor bcharacterizes the effectiveness of side y incombat. Thus, the equation1 dx2w··bshows that constant b measures the averageeffectiveness of each man on side y. Thesame interpretation can be given to the term——cx (t). Of course, there is no simple wayin which we can calculate the effectivenesscoefficients b and c. One way is to writethese coefficients in the formb == I',/py, C == Txpx, (48)

60 Differential Equations in Applicationswhere ry and rx are the coefficients of fire-power of sides y and x, respectively, and pyand px are the probabilities that each shotfired by sides y and x, respectively, provesaccurate.We note further that the terms correspond-ing to combat losses in the A-type systemare linear, whereas in the B-type systemthey are nonlinear. The explanation lies inthe following. Suppose that the guerillaforces amounting to x (t) personnel occupy acertain territory R and remain undetectedby the opposing side. And although the lat-ter controls this territory by firepower, itcannot know the effectiveness of its ac-tions. lt is also highly probable that thelosses suffered by the guerilla forces x are,on the one hand, proportional to the num-ber of personnel sc (t) on R and, on theother, to the number y (t) of personnel ofthe opposing side. Hence, the term corre-sponding to the losses suffered by the gueril-la forces x has the form —ga; (t) y (t), wherethe coefficient reflecting the effectiveness ofcombat operations of side y is, in general,more difficult to estimate than the coef-fiient b in the first relationship in (48).Nevertheless, to find g we can use the fire-power coefficient ry and also allow for theideas expounded by Lanchester, accordingto which the probability of an accurateshot fired by side y is directly proportionalto the so-called territorial effectiveness A,-y

Ch. 1. Construction of Differential Models 61of a single shot Bred by y and inverselyproportional to the area Ax of the territoryR occupied by side x. Here by ATU wedenote the area occupied by a single gueril-la frghter. Thus, with a high probabilitywe can assume that the formulas forfinding g and h areA Ag=ry-f, h=rx—x-{Ji-. (49)Below we discuss in greater detail each ofthe three differential models.Case A (differential systems of the A-typeand the quadratic law). Let us assume thatthe regular troops of the two opposingsides are in combat in the simple situationin which the losses not associated directlywith combat operations are nil. If, in addi-tion, neither side receives any reinforce-ment, the mathematical model is reducedto the following differential system:d déz-. -by, é: -cx. (50)Dividing the second equation by the first,we find thatdy __ isi-5 -- by . (51)Integrating, we arrive at the following re-lationship:b [y2 (t) -— y:] = c [x2 (t) — xg]. (52)

62 Differential Equations in Applicationsya,) A’>0, y winsK = 0) evenWl//b K < Q .z* wins0 r{/-—/(/c $(5)Fig. 12This relationship explains why system (50)corresponds to a model with a quadraticlaw. If by K we denote the constant quanti-ty by§ —— c:c§, the equationbyz —- cxz = K (53)obtained from Eq. (52) specifies a hyperbola(or a pair of straight lines if K = 0) andwe can classify system (50) with great pre-cision. We can call it a differential systemwith a hyperbolic law.Figure 12 depicts the hyperbolas for differ-ent values of K; for obvious reasons weconsider only the first quadrant (:1: > 0,y > 0). The arrows on the curves point in

Ch. 1. Construction of Differential Models 63the direction in which the number of per-sonnel changes with passage of time.To answer the question of who wins inthe constructed model (50), we agree to saythat side y (or ac) wins if it is the first towipe out the other side x (or y). For exam-ple, in our case the winner is side y ifK > 0 since in accordance with Eq. (53)the variable y can never vanish, while thevariable sc vanishes at y (t) = l/K/b. Thus,for y to win a victory, it must strive for asituation in which K is positive, that is,byg > c:1:§. (54)Using (48), we can rewrite (54) in the formyo 2 fx Px<..> > ry saThe left-hand side of (55) demonstrates thatchanges in the personnel ratio yo/xo givean advantage to one side, in accordancewith the quadratic law. For example, achange in the yo/.2:0 from one to two gives ya four—fold advantage. Note also thatEq. (53) denotes the relation between thepersonnel numbers of the opposing sides butdoes not depend explicitly on time. To de-rive formulas that would provide us with atime dependence, we proceed as follows.We differentiate the first equation in (50)with respect to time and then use the secondequation in this system. As a result we

64 Differential Equations in Applications·” .9%(Ui.M2•-•-—·-1--·-•-•—-Zv-•· *¤¤I1¥l¤Q0 ( .TFig. 13arrive at the following differential equation:dh:EF ‘·" bC.’IZ T-"‘ O.If for the initial conditions we takedx$(0)=%» ·g;L=0= —b1/0,then the solution to Eq. (56) can be ob-tained in the formsc (t) = :1:0 cos [St — yyo sin Bt, (57)where B = VE and Y =In a similar manner we can show thaty (t) L-. yo cos Bt — gg- sin Bt. (58)Figure 13 depicts the graphs of the func-tions specihed by Eqs. (57) and (58) in

Ch. 1. Construction of Differential Models 65the case where K >O (i.e. by§ > cxg, orYU 0 > xo)- _ _We note in conclusion that for side y towin a victory it is not necessary that yu begreater than xo. The only requirement isthat vyo be greater than xo.Case B (difierential systems of t.he B-typeand the linear law). The dynamical equa-tions that model combat operations betweentwo opposing sides can easily be solved,as in the previous case, if we exclude thepossibility of losses not associated withcombat and if neither side receives rein-forcements. Under these limitations theB—type differential system assumes the formd df-: ——g¤¤y, -5%: —h¤¤y- (59)Dividing the second equation in (59) bythe first, we get the equationi!!...£dx -_ g ’which after being integrated yieldsg ly (¢) —- ytl = h lx (¢) —- wo]- (60)This linear relationship explains why thenonlinear system (59) corresponds to a mo-del with a linear law for conducting combatoperations. Equation (60) can be rewrittenin the formgy — hx = L, (61)5—0770

66 Differential Equations in Applications.90} DQ _ywz'nsl=0, EUC/Z4/y L·<Q, .z* wirzs0 .·;//7 mw)Fig. 14with L = gy,) -— hay,. This implies, for one,that if L is positive, side y wins as a resultof combat operations, while if L is nega-tive, side sc wins.Figure 14 provides a geometrical inter-pretation of the linear law (61) for differentvalues of L.Let us now study in greater detail thesituation where one of the sides wins. Weassume that this is side y. Then, as we al-ready know, gyo — hxo must be positive, orL l$0 > 8 °If we now turn to (49), we see that side ywins if_§/L "xArxAxxo > ryA,.yAy ' ( )

Ch. 1. Construction of Differential Models 67Thus, the strategy of y is to make the ratioyo/xo as large as possible and the ratioAx/A y as small as possible. From the practi-cal standpoint it is more convenient towrite inequality (62) in the formAy!/0 ’°xArxAxxo > "yAry •We see that in a certain sense the productsA yyo and Axxo constitute critical values.We note, finally, that by combining(61) and (59) we can easily derive formulasthat give the time dependence of changesin the personnel numbers of both sides.Case C (differential systems of the C—typeand the parabolic law). In the C-model theguerilla forces face regular troops. We in-troduce the simplifying assumptions thatneither is supplied with reinforcements andthat the losses associated directly with com-bat operations are nil. In this case we havethe following system of differential equa-tions:where :2: (t) is the number of personnel inthe guerilla forces, and y (t) the number ofpersonnel in the regular troops. Dividingthe second equation in (63) by the first, wearrive at the differential equation12 -.;dx " gy '5*

68 Differential Equations in Applications_y(£) /V/>Q ywi/ZS /‘/=Q eve/z1//Qi, /*/<Q .z· wins·M/20 .z’(t')Fig. 15Integrating with the appropriate limitsyields the following relationship:gyz (t) =.= 20:1: (t) -§—M, (64)where M : gy§ -—- 2c;zv0. Thus, the systemof differential equations (63) corresponds toa model with a parabolic law for conductingcombat operations. The guerilla forces winif M is negative; they are defeated if M ispositive.Figure 15 depicts the parabolas definedvia Eq. (64) for different values of lll.Experience shows that regular troops candefeat guerilla forces only if the ratio yo/:::0considerably exceeds unity. Basing our rea-soning on the parabolic law for conductingcombat operations and assuming M posi-tive, we conclude that the victory of regular

Ch. 1. Construction of Differential Models 69troops is guaranteed if (yo/x0)2 is greaterthan (2c/g) .1:;*. If we allow for (48) and(49), we can rewrite this condition in theformy 2 T A p_” 1> 2 i ·y‘t .;;_1.12 Why Are Pendulum ClocksInaccurate?To answer this question, let us consder anidealized model of a pendulum clock con-sisting of a rod of length Z and a load of massm attached to the lower end of the rod(the rod’s mass is assumed so small that itcan be ignored in comparison to m) (Fig-ure 16). Iilthe load is;def1eoted by an angle ot7{ `XCX\\Ev \\I 8 \\\l x>| /t r'nzL·=—-rzFig. 16

70 Differential Equations in Applicationsand then released, in accordance withenergy conservation we obtainil§°i=mg(lcos9—lcosu), (65)where v is the speed at which the loadmoves, and g is the acceleration of gravity.Since s == l9, we have v = ds/dt ==l(d9/dt) and (65) leads us to tl1e followingdifferential equation:-g.(-‘§)2=g(e0Se—¤0Sa). (ee)If we now allow for the fact that -6 decreaseswith the passage of time t (for small fs),we can rewrite Eq_. (66) in the form `dt: —· I .1/ 23 ]/cos6—-—cosotIf we denote the period of the pendulumby T, we have_ oJ;- .. LS ...19...4 1/ 29 ]/cost)-cosoc ’(IorT Q d6I/28 0 ]/cosB·—cosoz ( )The last formula shows that the period ofthe pendulum depends on ot. This is the

Ch. 1. Construction of Differential Models 71main reason why pendulum clocks are inac-curate since, practically speaking, everytime that the load is deflected to the ex-treme position the deflection angle differsfrom cx.We note that formula (67) can be writtenin a simpler form. Indeed, sincecos9=l—2sin2g— , cosoc:-1-—-2 sin2E°2— ,we have___ GT- 2 I/L S ...........@._.......8 _ on _ 60 I/-sin? -2——s1n2 -5-"' °° d90 I/k2—si112 -5with k = sin (cx/2).Now instead of variable 9 we introducea new variable cp via the formulasin (6/2) = lc sin cp. This implies that when6 increases from O to ct, the variable cpgrows from O to rt/2, withgoes-;-d6=kcosq>dcp,or2 I/k’——sin’2k cos cp dcp 2COS? }/1-k’ s1n°q>

72 Differential Equations in ApplicationsThe last relationship enables us to rewriteformula (68) in the form[Ta/2 d.. ___ ....._‘L.....T"4l‘ g § ]/1—k2sin”q>:.4 1/.é.F(k, it/2),where the functionw dwF k, = S ————————-———( 0 1/1—k2 sinzcpis known as the elliptic integral of the firstkind, differing from the elliptic integral ofthe second kindwbE'(k, ip)-: S l/1——k2sin2q> dqn.0Elliptic integrals cannot be expressed interms of elementary functions. Therefore,all further discussion of the pendulum prob—lem will be related to an approach consid-ered when conservative systems are studiedin mechanics. Here we only note that thestarting point in our studies will be thedifferential equation-gag--f-kSiH9=O, k=which can be obtained from Eq. (60) bydifferentiating with respect to t.

Ch. 1. Construction of Differential Models 731.13 The Cycloidal ClockWe have established that clocks with ordi-nary (circular) pendulums are inaccurate.Is there any pendulum whose period is in-dependent of the swing? This problem wasfirst formulated and solved as early as the17th century. Below we give its solution,but first let us turn to the derivation of theequation of a remarkable curve, whichGalileo Galilei called the cycloid (from theGreek word for "circular"). This is a planecurve generated by a point on the circum-ference of a circle (called the generatingcircle) as it rolls along a straight line with-out slippage.Suppose that the x axis is the straightline along which the generating circle rollsand that the radius of this circle is r (Fig-ure 17). Let us also assume that initiallythe point that traces the cycloid is at theorigin and that after the circle turns throughan angle 6, the point occupies position M.Then, on the basis of geometrical reasoningwe obtainx = OS ·-= OP -— SP,y-=MS=CP——CN.ButOP =MP ==r9, SP =MN=rsin9,CP = r, CN = rcostl.

74 Differential Equations in Applications.9M L 10 A .z·_ 3 P ZJz'r· ` _Fig. 17Hence, parametrically the cycloid is spec_i-fied by the following equations:x=r(9-—sin9), y=r(1—cos6). (69)If we exclude parameter 9 from these equa-tions, we arrive at the following equationof a cycloid in a rectangular Cartesian co-ordinate system Oxy:33 =.—rcos·1(-K;-i)— l/2ry-—y2.The very method of constructing a cy-cloid implies that the cycloid consists of con-gruent arcs each of which corresponds to afull revolution of the generating circle.** The reader may find many interesting facts aboutthe cycloid and related curves in G.N. Berman’sbook The Cycloid (Moscow: Nauka, 1980) (inRussian).

Ch. 1. Construction of Differential Models 75The ‘s,epara_te arcs are linked at pointswhere they have a common vertical tan-gent. These points, known as cusps, corre-spond to the lowest possible positions occu-pied by the point on the generating circlethat describes the cycloid. The highestpossible positions occupied by the samepoint lie exactly in the middle of each areand are known as the vertices. The distancealong the straight line between successivecusps is 2:nr, and the segment of thatstraight line between successive cusps isknown as the base of an arc of the cycloid.The cycloid possesses the following prop-erties:(a) the area bounded by an are of a cycloidand the respective base is thrice the area ofthe generating circle (Galileo’s theorem);(b) the length of one cycloid arc is fourtimes the length of the diameter of the gen-erating circle (Wren’s theorem).The last result is quite unexpected, sinceto calculate the length of such a simplecurve as the circumference of a circle it isnecessary to introduce the irrational numberat, whose calculation is not very simple,while the length of an arc of a cycloid isexpressed as an integral multiple of the ra-dius (or diameter) of the generating circle.The cycloid has many other interestingproperties, which have proved to be extreme-ly important for physics and engineering.For example, the proiile of pinion teeth

76 Differential Equations in Applications:4 /1¥“ mléeiVFig. 18and of many typos of eccentrics, cams, andother mecha11ical parts of devices have theshape of a cycloid.Let us now turn to the problem thatenabled the Dutch physical scientist, astro-nomer, and mathematician Christian Huy-gens (1629-1695) to build an accurate clockin 1673. The problem consists of buildingin the vertical plane a curve for which thetime of descent of a heavy particle slidingwithout friction to a fixed horizontal lineis the same wherever the particle starts onthe curve (it is assumed that initially, attime t = to, the particle is at rest). Huy—gens found that the cycloid possesses suchan isochronous (from the Greek words for"equal" and "time") or tautochronousproperty (from tautos for "identical").From the practical standpoint the prob-lem can be solved in the following manner.Let us assume that a trough in the form of acycloid is cut out of a piece of wood asshown in Figure 18. We take a small metalball and send it rolling down the slope.Ignoring friction, let us try to determine

Ch. 1. Construction of Differential Models 77the time it takes the ball to reach the low-est possible point, K, starting from pointM, say.Let :1:0 and y0 be the coordinates of theinitial position of the ball, i.e. point M,and 90 the corresponding value of para-meter 9. When the ball reaches a pointN (9),the distance of descent along the verticalwill be h, which in view of Eq. (69) can befound in the following manner:h=y—y0=r(1——cos6)—— r (1 — cos 60) = r (cos 90 —— cos 9).We know that the speed of a falling objectis given by the formulaU= l/EEE,where g is the acceleration of gravity. Inour case the last formula assumes the formv.-: l/2gr (cos 90- cos 9)On the other hand, since speed is the deriv-ative of distance s with respect to time T,we arrive at the formulaTg-: V 2gr (cos 90 ——cos 6)Since for a cycloid ds = 2r sin (9/2) dB, wecan rewrite this formula (which is actuallya differential equation) in the formdT :2 2r sin (9/2) df}]/2gr (cos 90 — cos 6) ·

78 Diiierential Equations in ApplicationsIntegrating this equation within approp-riate limits, we obtainJTT__ S 2r sin (B/2) dBU `|/2gr(cos00-—cos0)___ TIS___2l/LS dc0s(9/2)g 0 ‘.·/ cos? -%°——-cos2g—== JTThus, the time interval T in the courseof which the ball rolls down from point Mto point K is given by the formulaT = TLwhich shows that period T is independentof 9,,, that is, the period does not dependon the initial position of the ball, M.Obviously, two balls that begin their mo-tion simultaneously from points M and Nwill roll down and find themselves at pointK at the same moment of time.Since we agreed to ignore friction, in itsmotion down the slope the ball passes pointK and, by inertia, continues its motionup the slope to point M1 lying at the samelevel as point M. Proceeding then in theopposite direction and traversing its path,the ball completes a full cycle. This con-stitutes the motion of a cycloidal pendulum

Cli. 1. Const.ruction of Differential Models 79\\\ \\ ~`\\ \\ `\\\` ///)”Fig. 19with a period of oscillationsT0 = lm Vr/g. (70)A peculiar feature of the cycloidal pen-dulum, which sets it apart from the simple(circular) pendulum, is that its period doesnot depend on the amplitude (or swing).Let us now show how an ordinary circu-lar pendulum can be made to move in atautochronous manner without resorting totrough and similar devices with consider-able friction. To this end it is sufficientto make a template (say, out of wood) con-sisting of two semiarcs of a cycloid withacommon cusp (Figure 19). The templateis fixed in the vertical plane and a stringwith a ball is suspended from the cusp O.The length of the string must be twice thediameter of the generating circle of thecycloid.If the ball on the string is deflected to anarbitrary point M, it begins to swing backg and forth with a period that is independ-

80 Differential Equations in Applicationsy ent of the positionof point M. Evenif due to frictiona11d air drag theamplitude of the\§_ oscillations dimin-u ishes, the period'° remains unchan-A ged. For a circu-B lar pendulum,C if- which moves alongan arc of a cir-Fig· 20 cumference,the iso-chronous proper-ty is satisfied approximately only forsmall amplitudes, when the arc differslittle from the arc of a cycloid.By way of an example let us study thesmall oscillations that a pendulum executesalong the arc AB of a cycloid (Figure 20).lf these oscillations are very small, theeffect of the guiding template is practicallynil and the pendulum oscillates almost likean ordinary circular pendulum with a string(the "rod") whose length is 4r. The path ABof a cycloidal pendulum will differ littlefrom the path CE of a circular pendulumwith a string length equal to [ir. Hence, theperiod of small oscillations of a circularpendulum with a string length Z: 4r ispractically the same as that of a cycloidalpendulum of the same length.

Ch. 1. Construction of Differential Models 81Now, if in (70) we put r == Z/4, the periodof small oscillations of a circular pendu-lum can be expressed in terms of the lengthof the string:T = 2% l/{EThis formula is derived (in a different way)in high school physics.In conclusion we note that the cycloidis the only curve for which a particle mov-ing along it performs isochronous oscilla-tions.1.14 The Brachistochrone ProblemThe problem concerning the brachistochrone(from the Greek words for "shortest" and"time"), a curve of fastest descent, was pro-posed by the Swiss mathematician JohnBernoulli (1667-1748) in 1696 as a challengeto mathematicians and consists of the fol-lowing.Take two points A and B lying in a ver-tical plane but not on a single vertical line(Figure 21). Among the various curves pass-ing through these two points we must {indA

Fig. 216-0170 .

82 Differential Equations in ApplicationsA U.al OQ| .2* n c-.z’I ’O | I1 ar 1i 1 ZI % :{ nC 15Fig. 22the one for which the time required for aparticle to fall from point A to point Balong the curve under the force of gravity isminimal.The problem was tackled by the bestmathematicians. It was solved by JohnBernoulli himself and also by Gottfried W.von Leibniz (1646-1716), Sir Isaac Newton(1642-1727), Guillaume L’Hospital (1661-1704), and Jakob Bernoulli (1654-1705).The problem is famous not only from thegeneral scientific viewpoint but also forbeing the source of ideas in a completelynew field of mathematics, the calculus ofvariations.Solution of the brachistochrone problemcan be linked with that of another problemoriginating in optics. Let us turn to Fig-ure 22, in which a ray of light is depicted aspropagating from point A to point P with

Ch. 1. Construction of Differential Models 83a velocity U1 and then from point P topoint B in a denser medium with a lower ve-locity v2. The total time Trequired for theray to propagate from point A to point Bcan, obviously, be found from the followingformula:T: 1/a¤v+x¤ _,r]/z»¤+;c-x)¤ l1 2If we assume that the ray of light propa-gates from pointA to point B along this pathin the shortest possible time T, the deriva-tive dT/dx must vanish. But thenI __ C--.23vi VPKF? vi 7orsincil __ sinoiz—Z""—?5I"The last formula expresses Snell’s well-known law of refraction, which initiallywas discovered in experiments in the formsin oil/sin oiz = a, with a constant.The above assumption that light choosesa path from A to B that would take theshortest possible time to travel is known asFermat’s principle, or the principle of leasttime. The importance of this principle liesnot only in the fact that it can be taken asa rational basis for deriving Snell’s law butalso, for one, in that it can be applied tofinding the path of a ray of light in a me-dium of variable density, where generally60

84 Differential Equations in ApplicationsZ4 lil ``—``—"``` "vg O, OE"` `````' "........ .1...2..- __ ____v GT..2... .... .. .... .1.05 _____v4 I1%Fig. 23the light travels not along straight—linesegments.For the sake of an example let us con-sider Figure 23, which depicts a ray of lightpropagating through a layered medium. Ineach layer the speed of light is constant,but it decreases when we pass from an up—per layer to a lower layer. The incident rayis refracted more and more strongly as itpasses from layer to layer and moves closerand closer to the vertical line. ApplyingSnell’s law to the interfaces between thelayers, we getsin cz.1 _ sinotz __ sin ons _ sin oa,U1 _ V2 _ va _ V4 ·Now let us assume that the layer thick-nesses decrease without limit while the num-ber of layers increases without limit. Then,in the limit, the velocity of light changes(decreases) continuously and we conclude

Ch. 1. Construction of Differential Models 85\ I\ OC~/1",.-.... -..-..i \| x\Fig. 24(soe Figure 24) that@2:% (7;)Uwith a = const. A similar situation is ob-served (with certain reservations) when aray of Sun light falls on Earth. As the raytravels through Eartl1’s atmosphere of in-creasing density, its velocity decreases andthe ray bends.Let us go back to the brachistochronoproblem. We introduce a system of coor-dinates in the vertical plane in the wayshown in Figure 25. We imagine that aball (like a ray of light propagating in me-dia) is capable of choosing the path of de-scent from point/1 to pointB with the short-est possible time of descent. Then, as fol-

86 Differential Equations in ApplicationsAI wn1I{Hx I\ nnI Ot \\\I \ BFig. 25lows from the above reasoning, formula (71)is valid.The energy conservation principle impliesthat the speed gained by the ball at agiven level depends only on the loss of po-tential energy as the ball reaches the leveland not on the shape of the trajectory fol-lowed. This means that v= l/2gy.On the other hand, geometric construc-tion enables us to show thatSlHG;COSB ZSGGB 1/1+ta¤¤|$....__L___1/4+<y*>2 °Combining the last two relationships with(71) yieldsE/[1 +(y')21== C· (72)

Ch. i. Construction of Differential Models 87This constitutes the brachistochrone equa-tion. We wish to demonstrate that only acycloid can represent a brachistochrone.Indeed, since y' = dy/dx, by dividing thevariables in Eq. (72) we arrive at the equa-tion—_ y l/2dx .- (————C__y ) dy.We introduce a new variable cp via thefollowing relationship:y */2...( C__y ) -tan cp.Theny=.-Csinz cp, dy=2C sinqpcoscpdcp,dx:-.tancpdy=2Csin2cpdcp= C (1 —— cos 2cp) dep.Integration of the last equation yieldssc == —g-(2cp—sin 2<p)—l—C,,where, in view of the initial conditions,:c=y=0 at cp:-0 and C1:-O. Thus,sz == —g (2cp—— sin 2q>),y=-Csin2cp=-g—(1—cos 2q>).Assuming that C/2 = r and 2rp = 9, wearrive at the standard parametric equations

88 Differential Equations in Applicationsof the cycloid (69). The cycloid is, indeed, aremarkable curve: it is not only isochro-nous——it is brachistochronous.1.15 The Arithmetic Mean, theGeometric Mean, and theAssociated Differential EquationLet us consider the following curious prob-lem first suggested by the famous Germanmathematician Carl F. Gauss (1777-1855).Let ml, and nl, be two arbitrary positivenumbers (ml, > 77.0). Out of ml, and nl, weconstruct two new numbers ml and nl thatare, respectively, the arithmetic mean andthe geometric mean of ml, and nl,. ln otherwords, we putmii , ni; Vm0n0•Treating ml and nl in the same manner asml, and nl,, we putmz :2%, nz: l/mlnl.Continuing this process indefinitely, we ar-rive at two sequences of real numbers, {mh}and {nh} (lc = O, 1, 2, .), which, as caneasily be demonstrated, are convergent. Wewish to know the difference of the limits ofthese two sequences.An elegant solution of this problemamounting to setting up a second-order lineardifferential equation is given below. It be-

Ch. 1. Construction of Differential Models 89longs to the German mathematician CarlW. Borchardt (1817-1880). Let a be thesought difference. It obviously depends onml, and nl,, a fact expressed analytically asfollows: cz =f (ml,, nl,),where f is a func-tion. The definition of a also implies thata =f(ml, nl). Now, if we multiply ml,and nl, by the same number k, each of thenumbers ml, 721, ml, nl, introducedabove, including a, will be multiplied byk. This means that a is a first—order homo-geneous function in ml, and nl, and, hence,a 2 mo}: (is no/mo) : m1]((1¤ nl/m1)'Introducing tlie notation x = nl,/ml,, xl =ni/my ·» U Z 1/lc (L no/m0)» yl =1/f (1, nl/ml), we find that__ m __ 2yly-- y1"`"£"""-`—-`—.1_l_x · (73)Since xl is related to sc via the equation__ 2 i/5$1* "W? iwe find thatdm _ 1—¤= __<¤¤r—=¤¥>(1+¢¤)”dn: _` (1+x)2l/Q _ 2(.r——x3) 'On the other hand, Eq. (73) leads to thefollowing relationship:dy __ ___ 2 2 dyl dxl§:-_` (1—l—:c)2 yl-l- 1-}-.2: dxl dw '

90 Differential Equations in ApplicationsReplacing dxl/dx with its value given bythe previous relationship and factoring out(x — x3), we {ind thatdy_2:v(x—i)<¤¤-x°> zs··wr;r·y1d+<4+x><=»l—¤¤$>§,§.Difierentiating both sides of this equationwith respect to x, we {ind thatd [<¤—r>%]dx[ x(a:—1) ]_2 d 1+x +2** ($*1) dl/1 d·‘¤1*“ yi dx 1+x dxl dxd+<xi—~¤i>—$d d ddx]. (1:;:1 'By elementary transformation this equa-tion can be reduced tod dyx[<x·x3>aal—¤*y_ i—x _d_ ___ _3 dy; __“(1+x) {dx1[(xi 11) dxl] xiyi} •If we now replace x with xl, then xl willbecome xg. If we then replace xl with xg,

Ch. 1. Construction of Differential Models 91we find that :::2 is transformed into x3, etc.Hence, assuming thatd dy ..g;,,—[(¤¤—-wa) ;,;]—<¤y-a* (y>»we arrive at the following formula:as, (y): 1--as 1--.1: 1——x2<1+¤=>1/5Z<<¤+¤·=i> V?{(i+¢2> Vi?1-:1:,, *>< >< —————-—; a y ).<1+xn>1/xi (”If we now send rz to infinity, we find that1 —- x,, tends to zero and, hence,¢* (y) = 0-This means that y satisfies the differentialequation(:1: —-— .2:3) -3.752) —%‘Q;;—xy=O. (74)If we note that2¤=r<m., mi)-=y —-—---’ ‘”,§;·· "··’ .0we can easily find the value of this num-ber. Indeed, since y must be the constantsolution to Eq. (74), we find that onlyy E O is such a solution. Thus, the differ-ence of the limits of the sequences {mn} and{nk} is zero.The differential equation (74) is remark-able not only because it enabled reducing

92 Differential Equations in Applicationsthe initial problem to an obvious one butalso because it is linked directly to the so-lution of the problem of calculating the pe-riod of small oscillations of a circular pen-dulum.As demonstrated earlier, the period ofsmall oscillations of a circular pendulumcan be found from the formulaT = 4]/UQF (lc, at/2),witharr/2 dF ky 2 —-T S ,( n/) 0 l/1—k2sin2cpIt has been found that if 0 { lc < 1, thensr/2S ·········—————“”`°0 1/1-kzsinzqn__:m ·’ OO i2><32><5”><...><(2n—1)2——?l1—l-E1 2°X42><6°X ...><(2f1)“ k2n)’1],:*.where_ OO 12><32><52><...><(2n—1)2 ny-1+21 22><4”><62>< X(2”)2 Z27],:*.is a solution to the differential equation (74).

Ch. 1. Construction of Differential Models 931.16 On the Flight of an ObjectThrown at an Angle to the HorizonSuppose that an object is thrown at an an-gle cx to the horizon with an initial velocityvo. We wish to derive the equation of theobject’s motion that ignores forces offriction (air drag).Let us select the coordinate axes as shownin Figure 26. At an arbitrary point of thetrajectory only the force of gravity Pequal to mg, with m the mass of the objectand g the acceleration of gravity, acts onthe object. Hence, in accordance withNewton’s second law, we can write thedifferential equations of the motion of theobject as projected on the :4: and y axes asfollows:d2 d2mq;-:0, m%:——mg.Factoring out m yieldsdaz ___ day _w—O· W- *5* (75)y __.,·:=;.“Et="mvva tm.?0 A .7.Fig. 26

94 Differential Equations in ApplicationsThe initial conditions imposed on the ob-ject’s motion are.CC=O, y=—-0, ·—?;:·-UOCOSOC,tTgzvosinoz. at t-—=O.Integrating Eqs. (75) and allowing for theinitial conditions (76), we find that theequations of the object’s motion aresc = (vo cos ot) 23, y = (vo sin cx) t — gtz/2.(77)A number of conclusions concerning thecharacter of the object’s motion can bedrawn from Eqs. (77). For example, we canfind the time of the object’s flight up to themoment when the object hits Earth, the rangeof the flight, the maximum height that theobject reaches in its flight, and the shapeof the trajectory.The first problem can be solved by findingthe value of time t at which y = 0. Thesecond equation in (77) implies that thishappens whent[v0 since--€§]=O,that is, either t = O or t = (2120 sin oc)/g.The second value provides the solution.The second problem can be solved bycalculating the value of x at a value of tequal to the time of flight. The first equa-

Ch. 1. Construction of Differential Models 95tion in (77) implies that the range of theflight is given by the formula(vo cos ct) (2v,, sin oc) _ 1:% sin 20:.8* ___ E °This implies, for one thing, that the rangeis greatest when 2ot == 90°, or ot = !i5° Inthis case the range is vg/g.The solution to the third problem can beobtained immediately by formulating themaximum condition for y. But this meansthat at the point where y is maximal thederivative dy/dt vanishes. Noting thatfg-: —·gt—|—v0 sin oc,we arrive at the equation -—gt + vo sin oc =O,which yieldst = (vo sin ot)/y.Substituting this value of t into the secondequation in (77), we find that the maximumheight reached by the object is v§ sinz on/2g.The solution to the fourth problem hasalready been found. Namely, the trajectoryis represented by a parabola since Eqs. (77)represent parametrically a parabola, whichin rectangular Cartesian coordinates canbe written as follows:y= x tan oc- gi? seczct.

96 Differential Equations in Applications1.17 WeightlessnessThe state of weightlessness (zero g) can beachieved in various ways, although it isassociated (consciously or subconsciously)with the “floating" of astronauts in thecabin of a spacecraft.Let us consider the following problem.Suppose that a person of weight P is stand-ing in an elevator that is moving down-ward with an acceleration co = osg, withO<ot<1 and g the acceleration of grav-ity. Let us determine the pressure thatthe person exerts on the cabin’s floor andthe acceleration that the elevator mustundergo so that this pressure will vanish.Two forces act on the person in the eleva-tor (Figure 27): the force of gravity P andthe force Q that the floor exerts on the per-QP.2*Fig. 27

Ch. 1. Construction of Differential Models 97son (equal numerically to the force of pres-sure of the person on the floor). The differ-ential equation of the person’s motion canbe written in the formdzxmq?--P—Q. (78)Since d20:/dtz =-— to = ug and m = P/g, wecan rewrite Eq. (78) as follows:d2Q:.-P—mTi£-:P(1—ot). (79)Since 0 < ot < 1, we conclude that Q < P.Thus, the pressure that the person exerts onthe floor of the cabin of an elevator mov-ing downward is determined by the forceQ = P (l — OL).On the other hand, when the elevator is mov-ing upward with an acceleration to = osg,O < oc < 1, the pressure that the personexerts on the floor of the cabin is deter-mined by the force Q = P (1 -{— ot). Let usnow establish at what acceleration the pres-sure vanishes. For this it is sufficient to putQ = O in (79). We conclude that in thiscase ot = 1, that is, for Q to vanish theacceleration of the elevator must be equalto the acceleration of gravity.Thus, when the cabin is falling freelywith an acceleration equal to g, the pres-sure that the person exerts on the floor isnil. lt is this state that is called weightless-7-0770

98 Differential Equations in Applicationsness. In it the various parts of a person’sbody exert no pressure on each other, sothat the person experiences extraordinarysensations. In the state of weightlessnessall points of an object experience the sameacceleration.Of course, weightlessness is experiencednot only during a free fall in an elevator.For illustration let us consider the followingproblem.What must be the speed of a spacecraftmoving around the Earth as an artificialsatellite for a person inside it to be in thestate of weightlessness?One assumption in this problem is thatthe spacecraft follows a circular orbit ofradius r + h, where r is Earth’s radius, andh is the altitude at which the spacecrafttravels (reckoned from Earth’s surface). Theprevious problem implies that in the stateof weightlessness the pressure on the wallsof the spacecraft is zero, so that the force Qacting on an object inside the spacecraftis zero, too. Hence, Q == O. Let us nowturn our attention to Figure 28. The as axisis directed along the principal normal rzto the circular trajectory of the spacecraft.We use the differential equation of the mo-tion of a particle as projected on the prin-cipal normal:TLmvzT: E Frm,h=l

Ch. 1. Construction of Differential Models 99Q’” (F—¤H $K.2:Fig. 28Tl.where p :r—\—h, E FM =-F, and F is1;:1directed along the principal normal to thetrajectory. We getmv? dz.1:T;@‘·F : mrmnor the equationdzx __ v2dig — r-|—hSubstituting this value of dzx/dig intoEq. (78), we find thatmv?—7_§-{-:1)-—Q. (80)

100 Differential Equations in ApplicationsHere force P is equal to the force F ofattraction to Earth, which, in accordancewith Newton’s law of gravitation, is inverse-ly proportional to the square of the dis-tance r -\~h from the center of Earth,that is,kmF " (r+h)2 ’where m is the mass of the spacecraft, andconstant lc can be determined from thefollowing considerations. At Earth’s sur-face, where h = 0, the force of gravity Fis equal to mg. The above formula thenyields lc = gr? Hence,p Z F: Jeri.(r+h>” ’where g is the acceleration of gravity atEarth’s surface.lf we now substitute the obtained valueof P into (80) and note that Q = 0, we findthat the required speed is given by theformula-- · 8U "‘ " l/ r+h1.18 Kepler’s Laws of PlanetaryMotionIn accordance with Newton’s law of gravi-tation, any two objects separated by a dis-tance r and having masses rn and M are at-

Ch. 1. Construction of Differential Models 101y \\V/*005 gv mI \ _F ___ FSL/7§0// Iw'/ ry/ IM / I70 .2* P arFig. 29tracted with a forceGmMF :71 (81)where G is the gravitational constant.Basing ourselves on this law, let us de-scribe the motion ofthe planets in the solarsystem assuming that m is the mass of aplanet orbiting the Sun and M is theSun’s mass. The effect of other planets onthis motion will be ignored.Let the Sun be at the origin of the coor-dinate system depicted in Figure 29 andthe planet be at time t at a point withrunning coordinates x and y. The attractiveforce F acting on the planet can be decom-posed into two components: one parallel tothe :1: axis and equal to F cos cp and theother parallel to the y axis and equal to

102 Differential Equations in ApplicationsF sin rp. Using formula (81) and Newton’ssecond law, we arrive at the followingequations:most: —-Fcos cp = —£?¥cos cp, (82)my: —Fsinq>:—§I}%sin cp. (83)Bearing in mind that sin cp = y/r andcos cp = .2:/r, we can rewrite Eqs. (82) and(83) as follows:•• k •• kx Z — “°;E;;` ¤ U Z _ jig/“ »where constant It is equal to GM.Finally, allowing for the fact that r =l/x2 -I— y2, we arrive at the differentialequations•• T- •• ix-_ — (x2_|_y2)3/2 9 y __' l-— (x2_*_y2)3/2 °(84)Without loss of generality we can assumethatx:a,y=—-O,x•=O,g)=v0a‘tt=O. (85)We see that the problem has been re-duced to the study of Eq. (84) under theinitial conditions (85). The special fea-tures of Eqs. (84) suggest that the most con-venient coordinate system here is the oneusing polar coordinates: x = r cos cp and

Ch. 1. Construction of Diiierential Models 103y = r sin q>. Thendzzrooscp — (rsinrp) tp,Q: rsincp + (rcoscp)q>,QJ: rooos cp - 2 ($— sm tp) gb (se)— (r sin cp) ·q; — (r oos cp) qlz,[;=•I:SlI1 cp —|— 2 cos cp)+ (r cos cp) — (r sin cp) qaz.Hence,dz.: —— rlpz) eos qu —— (2rtp —{— sin cp,Q.; (l’•—— Tq32)SlI1 (P + + COS (P.Using the last two relationships, we canrewrite the differential equations (84) inthe form(or.-- rep?) cos cp -— (Zrrp + sin cp= <87>(•7:—rcl>2) sin cp -|- (Zrco-I- cos rpZ .- . (88)Multiplying both sides of Eq. (87) bycos cp, both sides of Eq. (88) by sin ep, and

104 Differential Equations in Applicationsadding the products, we find thatro- rqlz = -k/rz. (89)Multiplying both sides of Eq. (87) bysin cp, both sides of Eq. (88) by cos cp, andsubtracting the second product from thefirst, we arrive at the equation2rqi + riii Z 0. (eo)As for the initial conditions (85), in polarcoordinates they assume the form(91)Thus, we have reduced the problem ofstudying Eqs. (84) under the initial condi-tions (85) to that of studying Eqs. (89) and(90) under the initial conditions (91). Wealso note that Eq. (90) can be rewritten inthe formd •-H-— (rzqn) = 0. (92)But Eq. (92) yieldsrah - C., <93>where C1 is a constant possessing an in-teresting geometric interpretation. Precise-ly, suppose that an object moves frompoint P to point Q along the arc PQ (Fig-ure 30). Let S be the area of the sector limit-ed by the segments OP and OQ and the

Ch. 1. Construction of Differential Models 1053Cl{0P .2Fig. 30arc From the calculus course we knowtl1atcpS--—i— rzd... 2 (P,0or dS = (1/2) rz dcp. Hence,dS _1 2 dcp _ 1 2’7F* 2 " at —?r "’• (94)The derivative dS/dt constitutes what isknown as the areal velocity, and since inview of (93) rzqn is a constant, we concludethat the areal velocity is a constant, too.But this, in turn, means that the objectmoves in such a manner that the radiusvector describes equal areas in equal timeintervals. This law of areas constitutes oneof the three Kepler laws. In full it can beformulated as follows: each planet movesalong a plane curve around the Sun in sucha manner that the radius vector connecting

106 Differential Equations in Applicationsthe S un with the planet describes equal areasin equal time intervals.To derive the next Kepler law, whichdeals with the shape of the planetary tra-jectories, we return to Eqs. (89) and (90)with the initial conditions (91) imposed onthem. The initial conditions imply, forone, that r = a and cp : vo/a at t = O.But then condition (93) implies that C1 =avo. Hence,rzcp = avo, or rp = avg/rz. (95)This transforms Eq. (89) into°‘ a2v2 lc':‘%’"Tr·Assuming that : p, we can rewrite thisequation in the form£-§rE;- Q- evt _ kdt_dr dt _p dr—_ r3 —F’ordp __ a2v§ _ kPw"—_r WSeparating the variables in the last diffe-rential equation and integrating, we arriveat the following relationship:TM r w+C2·Since p = : O at r : a, we {ind thatv2 lcC-‘¤··ij‘“T·

Ch. 1. Construction of Differential Models 107Thus, we arrive at the equationLz ____ k azvz v§ lcT- T r 22%+*: ‘ tvor, if we consider only the positive valueof the square root,dr ___ l/ 2k 2]:: a2v2ar- ("<·“T)+T”··;§“ (99Dividing Eq. (96) by Eq. (95), we hud that—§%·-—"L`7° -I/.(XJ°2···l·2Br···1,where__ 1 ___ 2k B_ lcQ--? a3v§ ’ — a2v§,The last equation can be integrated by sub-stituting 1/u for r. The result isT: a,2v§/lc1+¤<>¤S(<¤>+6’3> ’where e = il/cx + [32/B = avg/lc —-— 1. Theconstant C3 can be determined from thecondition that r = a at cp = O. It is easyto verify that C3 = O. Thus, we finally have_ a2v§/lcrv 1+ecos<p (97)From analytic geometry we know that thisis the equation of a conic in terms of polarcoordinates, with e the eccentricity of theconic. The following cases are possible

108 Differential Equations in Applicationshere:(1) an ellipse if e<1, or vg<2k/cz,(2) a hyperbola if e > 1, or vg > 2k/a,(3) a parabola if e : 1, or vg = 2k/a,(4) a circle if e = O, or vg = lc/o.Astronomical observations have shownthat for all the planets belonging to thesolar system the value of vg is smallerhan 2l.c/a. We, therefore, arrive at anotherof Kepler’s laws: the planets describe el-lipses with the Sun at one focus.Note that the orbits of the Moon andthe artificial satellites of Earth are alsoellipses, but in the majority of cases theseellipses are close to circles, that is, e differslittle from zero.As for recurrent comets, like, say, Hal-ley’s comet, their orbits resemble "prolate"ellipses whose eccentricity is smaller thanunity but very close to it. Say, Halley’scomet appears in Earth’s neighborhoodapproximately every 76 years. Its latestapparition was in the period between theend of 1985 and the beginning of 1986.Celestial bodies that move along para-bolic and hyperbolic orbits may be observedonly once, since they never return tothe same place.Let us now establish the physical mean-ing of eccentricity e. First we note thatthe components x and y of a planet’s vel-ocity vector Valong the as and y axes,_respec-

Ch. 1. Construction of Differential Models 109tively, and the size v of vector V satisfythe relationshipU2 I WL Q2,which when combined with (86) yieldsw:#&+hFrom this it follows that the kinetic energyof a planet of mass m is given by the for-mula% mvz = é- m (rzcpz + rz). (98)The potential energy of a system is minusone times the amount of work needed tomove the planet to infinity (where the po-tential energy is zero by convention).Hence,OO km km <>¤ kmTlf by E we denote the total energy of thesystem, which in view of energy conserva-tion must be constant, then formulas (98)and (99) yield1 2.2 °2 kmAssuming that cp = 0 and combining (97)with (100), we get__ azvz/lc mr2a2v2 _ km __'·· ¢‘··E·

110 Differential Equations in ApplicationsExcluding r from the last two relationships,we {ind thatP 2a2v§" Z l/1 +L nmEquation (97) for the shape of the orbitfinally assumes the formT: azvg/lc1+ ]/1—{—E (2a2v§/mkz) cos cpThis formula implies that the orbit is anellipse, hyperboba, parabola, or circle if,respectively, E < O, E >O, E = O, orE = —mk2/2a,2v§. Thus, the shape of aplanet’s orbit is completely determined bythe value of E. Say, if we could impartsuch a "blow" to Earth that it would increaseEarth’s total energy to a positive value,Earth would go over to a hyperbolic orbitand leave the solar system forever.Let us now turn to Kepler’s third law.This law deals with the period of revolu-tion of planets around the Sun. Takinginto account the results obtained in deriv-ing the previous Kepler law, we naturallyrestrict our discussion to the case of ellip-tic orbits, whose equation in terms of Carte-sian coordinates, as is known, is:1:2 12@+%:1*

Ch. 1. Construction of Differential Models 111.7AC F J2g IY' {Fig. 31where (Figure 31) the eccentricity e = C/E,with C2 = E,2 —— n2, so thate2 ·—= (E2 — n2)/E2,orn2 = E2 (1 —— B2)- (101)Combining this with (97) and allowing forthe properties of an ellipse, we concludethat___ 1 a2v2/k a2v2/k __ a2v21-lge + 1-3:2 )_lc(1-0422)¤2v3E”I M12 vornz: 1%%.*; _ (102)

442 Differential Equations in ApplicationsLet us denote the period of revolution of aplanet by T. By definition, T is the timeit takes the planet to complete one full or-bit about the Sun. Then, since the area lim-ited by an ellipse is Jtgn, we conclude, onthe basis of (94) and (95), that at§n ==av0T/2. Finally, taking into account (102),we arrive at the following result:4;-{,2§2»n2 43-[2T2: —.a.:g‘—:T€“·This constitutes the analytical descriptionof Kepler’s third law: the squares of theperiods of revolution of the planets are pro—portional to the cubes of the major axes of theplanets’ orbits.1.19 Beam DeflectionLet us consider a horizontal beam AB(Figure 32) of a constant cross section andmade of a homogeneous material. Thesymmetry axis of the beam is indicated inFigure 32 by a dashed line. Suppose thatforces acting on the beam in the verticalplane containing the symmetry axis bendthe beam as shown in Figure 33. These forcesmay be the weight of the beam itself oran external force or the two forces actingsimultaneously. Clearly, the symmetry axiswill also bend due to the action of theseforces. Usually a bent symmetry axis iscalled the elastic line. The problem of deter-

Ch. 1. Construction of Differential Models 113mining the shape of this line plays an im-portant role in elasticity theory.Note that there can be various types ofbeam depending on the way in which beamsare fixed or supported. For example, Fig-ure 34 depicts a beam whose end A is rigidlyfixed and end B is free. Such a beam is saidto be a cantilever. Figure 35 depicts a beamlying freely on supports A and B. Anothertype of beam with supports is shown inFigure 36. There are also various ways inwhich loads can be applied to beams. Forexample, a uniformly distributed load isshown in Figure 34. Of course, the load canvary along the entire length of the beamor only a part of this length (Figure 35).Figure 36 illustrates the case of a concen-trated load.Let us consider a horizontal beam OA(Figure 37). Suppose that its symmetryaxis (the dashed line in Figure 37) lies onthe :1: axis, with the positive directionbeing to the right of point O, the origin.The positive direction of the y axis isdownward from point O. External forcesF1, F2, (and the weight of the beamif this is great) bend the symmetry axis,which becomes the elastic line (also depict-ed in Figure 38 by a curved dashed line).The deflection y of the elastic line from theIII axis is known as the sag of the beam atpoint x. Thus, if we know the equation ofthe elastic line, we can always find the sag8-0770

114 Differential Equations in ApplicationsA £m/# m#//1///0/////100// BFig. 32A B%”""Ill§!.."’”/MFig. 33Al B§§Fig. 34of the beam. Below we show how this canbe done in practical terms.Let us denote by M (x) the bending mo-ment in the cross section of the beam atcoordinate .2:. The bending moment is de-fined as the algebraic sum of the momentsof the forces that act from one side of thebeam at point x. In calculating the mo-ments we assume that the forces acting onthe beam upward result in negative mo-ments while those acting downward resultin positive moments.

Ch, 1. Construction of Diherential Models 145P czFig. 35_ PA BsFig. 36F F M1 fg 5 (Z`) G0 Aa:JI}.9Fig. 37The strength-of—materials course provesthat the bending moment at point x is re-lated to the curvature radius of the elasticHDB via HIE 9ql1&tiOIlEJ .4.:M , 103u+<y >=1=··x= (x) ( )where E is Young’s modulus, which de-8*

116 Differential Equations in Applications5F0 3 fl;wl F5 W`4y (DFig. 38pends on the type of material of the beam,J is the moment of inertia of the cross sec-tion of the beam at point as about the hori-zonta-1 straight line passing through thecenter of mass of the cross section. Theproduct EJ is commonly known as theflexural rigidity, in what follows we assumethis product constant.Now, if we suppose that the sag of thebeam is small, which is usually the case inpractice, the slope y' of the elastic line isextremely small and, therefore, instead ofEq. (103) we can take the approximateequationEJ y" = M (:1:). (104)To illustrate how Eq. (104) is used inpractice, we consider the following prob-lem. A horizontal homogeneous steel beamof length l lies freely on two supports and

Ch. 1. Construction of Differential Models 117pi/2 pz/2L-a:0 X A—` __________ ,....--··’ .2:1 Q I9 *°°` ,¤<&·~v}Fig. 39sags under its own weight, which is p kgfper unit length. We wish to determine theequation of the elastic line and the maxi-mal sag.In Figure 39 the elastic line is depictedby the dashed curve. Since we are dealingwith a two-support beam, each support actson the beam with an upward reaction forceequal to half the weight of the beam (orpl/2). The bending moment M (sc) is thealgebraic sum of the moments of these forcesacting on one side from point Q (Fig-ure 39). Let us first consider the action offorces to the left of point Q. At a distance xfrom point Q a force equal to pl/2 acts onthe beam upward and generates a negativemoment. On the other hand, a force equalto px acts on the beam downward at adistance .1/2 from point Q and generates apositive moment. Thus, the net bending

118 Differential Equations in Applicationsmoment at point Q is given by the formulaz 2 1Mw: ···§Lx+Px(%)=f%·—%£(105)If we consider the forces acting to theright of point Q, we find that a force equalto p (l — :1;) acts on the beam downward ata distance (l -- sz:)/2 from point Q and gen-erates a positive moment, while a forceequal to pl/2 acts on the beam upward at adistance l — az from point Q and generatesa negative moment. The net bending mo-ment in this case is calculated by the for-mulaM(x>=·p (i——¤¤>i?—§L<i—x>..r.2=i-E2... 2 .2 (106)Formulas (105) and (106) show that thebending- moments prove to be equal. Now,knowing how to find a bending moment, wecan easily write the basic equation (104),which in our case assumes the formEly": Jl? -—%E (107)Since the beam does not bend at points Oand A, we write the boundary conditionsfor Eq. (107) so as to {ind y:y=Oatx=Oand.py=O,atx:l. ..

Ch. 1. Construction of Differential Models 119Integration of Eq. (107) then yieldsy = @1% (xi -— 2Zx3—|— Pcs). (108)This constitutes the equation of the elasticline. It is used to calculate the maximalsag. For instance, in this example, basingour reasoning on symmetry considerations(the same can be done via direct calcula-tions), we conclude that the maximal sagwill occur at :2: = Z/2 and is equal to5pl‘*/384E./`, with E =-— 21 >< 105 kgf/cm2and J = 3 >< 104 cm4.1.20 Transportation of LogsIn transporting logs to saw mills, loggingtrucks move along forest roads some of thetime. The width of the forest road is usual-ly such that only one truck can travel alongthe road.Sections of the road are made wid-er so that trucks can pass each other. Ignor-ing the question of how traffic should bearranged that loaded and empty traffictrucks meet only at such sections, let usestablish how wide the turns in the roadmust be and what trajectory the drivermust try to follow so that, say, thirty-me-ter logs can be transported. It is assumedthat the truck is sufficiently maneuverableto cope with a limited section of the road.Usually a logging truck consists of atractor unit and a trailer connected freely

120 Differential Equations in ApplicationsS ,9 C Q PX B A Ys R Q P A. FFig. 40to each other. The tractor unit has a front(driving) axle and two back axles abovewhich a round platform carrying two postsand a rotating beam are fastened. Thisplatform can rotate in the horizontal planeabout a symmetrically positioned verticalaxis. One end of each log is placed on thisplatform. The trailer has only two backaxles, but also has a platform with twoposts. The other ends of the logs are put onthis platform. The trailer’s chassis consistsof two metal cylinders one of which canpartly move inside the other. The chassisconnects the back platform with the axisthat links the trailer with the tractor unit.Thus, the length of the chassis can changeduring motion, which enables the tractorunit and the trailer to move independentlyto a certain extent. Schematically the log-ging truck is depicted in Figure 40. Thepoints A and B correspond to the axes ofthe platforms a distance h apart. By XYwe denote a log for which AX : kh. PointC corresponds to a small axis that connects

Ch. 1. Construction of Differential Models 121the tractor unit with the trailer, withAC=ah. Usually a-:0.3, but in thesimplest case of log transportation a = O.Next, FF is the front axle of the tractorunit, and PP and QQ are the back axles ofthe tractor unit, while RR and SS arethe trailer axles. All axles have the samelength 2L, so that for the sake of simplic-ity we assume that the width of the log-ging truck is 2L. As for the width of theload in its rear, we put it equal to 2W.In what follows we will need the conceptof the sweep of the logging truck, which iscommonly understood to be the maximumdeflection of the rear part of the loggingtruck (for the sake of simplicity we assumethis part to be point X) from the trajectoryalong which the logging truck moves. Letus suppose that the road has a width of2[iih and that usually a turn in the road issimply an arc of a circle of radius h/czcentered at point O (Figure 41). For thesake of simplicity we assume that a loggingtruck enters a turn in the road in such away that the tractor unit and the trailerare positioned along a single straight line,the driver operates the truck in such a waythat point A, corresponding to the axis ofthe front platform, is exactly above theroad’s center line. Point A in Figure 41is determined by the angle X that the truckAC makes with the initial direction. Hereit is convenient to fix a coordinate system

122 Differential Equations in ApplicationsX\\\\Zrzifial __ g_8 \_\____dirccfian '/’ \\// \/ 0xxy // // A/// //// \// /`/// \/ /.// \x /’ ‘0 .z·Fig. 41Oxy in such a way that the horizontal axispoints in the initial direction and the ver-tical axis is perpendicular to it. In a generalsituation the load carried by the truckwill make an angle 6 with the initial direc-tion. As for angle BAC in Figure 41,denoting it by u we {ind that u, = X —— 9.Usually this angle is the logging truck’sangle of lag. The required halfwidth h ofthe road, which determines the sweep of thelogging truck at a turn and is known as thehalfwidth of the road at the outer curb of a

Ch. 1. Construction of Differential Models 123turn, is defined as the algebraic sumOX — OA —l— W, while the halfwidth of theroad at the inner curb of a turn is definedas the algebraic sum OA + L — OP, whereOP is the perpendicular dropped frompoint O onto AB.We stipulate that in its motion a loggingtruck’s wheel either experiences no lateralskidding at all or the skidding is small.This requirement, for one thing, means thatthe center line AC of the tractor unit con-stitutes a tangent to the arc of a circle atpoint A, so that OA is perpendicular to AC,and angle X is determined by the motion ofpoint A along the arc of the circle. Note,further, that in building a road the curva-ture of a turn in the road is determined byan angle N° that corresponds to an arclength of a turn of approximately 30 m.In our notation,O i80° 30uN · ;—.—:T—T- , (109)where h is measured in meters. Thus, ath = 9 m and ot -:0.1 we have N° z 19°,while at h = 12 m and cx = 1.0 we haveN ° :=¢ 142°. For practical considerations wemust consider only such cx’s that lie be-tween 0 and 1, and the greater the value ofoc, the greater the maneuverability of thelogging truck.The log length 7th will be greater than h,but again, reasoning on practical grounds,

124 Differential Equations in Applicationsit must not be greater than 3h. Thus, thevalue of 2. varies between 1 and 3. As forconstant a, it is assumed that 0 { a < 0.5.Finally, we note that in each case thevalue of h is chosen differently, but it variesbetween 9 and 12 m.Since the wheels of the tractor unit donot skid laterally, the coordinates of pointA in Figure 41 arex= —E·sinX, y:-.£cosX.a exThe coordinates of point B areX=—;—sin X—hcos 9,Y: —£—cosX +hsin0. (110)Since the trailer’s wheels do not skid either,point B moves in the direction BC anddY-ai-—..:—tanxp, (111)where mp is the angle which BC makes withthe initial direction. Next, employing thefact that X = u. + 0 and studying triangleABC, we arrive at the following chain ofequalities:sin (X—1|>) ___ sin (0—1|>) __ sin u.‘—‘r"*———;»T*-Th · (**2)where 0<b<1, and xp, 0, and u are

Ch. 1. Construction of Differential Models 125functions of X. If we combine (111) with(110), we find that(—-Q-S1nx+z, %3—cos9)cos1pIt dB . .—{-(-Ex-cosX+hT1T(-s1n9)s1n1p-O.Carrying out the necessary calculations, W9arrive atsin(X——1p)== on —g§—cos(9—1p), (113)If we exclude variable 142 from this relation-ship fora fixed value of a by employing (112)and the fact that X =-· u —l— B, we arrive(since 9 : 0 at X = 0) at the differentialequation2.- -...22;... 114dx _1 0t(1——ac0Su) ( )with the initial condition u (0) :.· O, wherethe angle of lag plays the role of the soughtfunction.* _By substituting v for tan (u/2) in thedifferential equation (114), we can mtegratethe equation in closed form. However, theresulting u vs. X dependence proves to beextremely complicated, which, of course,hinders an effective study. Nevertheless,Eq. (114) can easily be integrated andstudied numerically. To this end we can* See A.B. Tayler, "The sweep of a logging truck",Math. Spectrum 7, No. 1: 19-26 (1974/75).

126 Differential Equations in Applicationsemploy, for example, the Runge—Kutta sec-ond—order numerical method the essence ofwhich is the following.Suppose that an integral curve u = cp (X)of a differential equation du/dX = j (X, u)passes through a point (X0, uc). At equidis-tant points xm xp xw ·» (xm —~ xi =AX >0) we select values uc, ul, u2, .such that ui x cp, (X), where the successivevalues ul, uz, are specified by theformulasui+1 Z ui ’l’ (ki ’l` k2)/2¤withki = f (xi, ui) AX»k2 : ft (Xi JF AX» ui ‘l‘ ki) AX-To solve Eq. (114) numerically with theinitial condition u (0) == 0, we compile thefollowing program using BASIC:10 REM Runge-Kutta second-order method20 DEF FNF (X, U) =1 — SIN (U)/(AL=•·(1 — A=•·COS(U)))30 CLS: PRINT "Soluti0n of differential equa-tion"40 PRINT "Runge—Kutta second-order method"100 PRINT "Parameters:"110 INPUT "Alpha=", AL120 INPUT "a=", A130 INPUT "Step of independent V&I'I&bI€=”, DX140 PRINT "Initial values:"150 INPUT "of independent V3.I‘I&bI6==·”, X

Ch. 1. Construction of Differential Models 127160 INPUT “of function=", U200 BEM Next 20 values of function210 CLS: PRINT "X", “U"220 FOR I :1 TO 20230 PRINT X, U240 K1 : DX»¤·FNF (X,U)250 X = X "l‘ DX260 K2 = DX*FNF(X, U -{— K1)270 U = U —|— (K1 + K2)/2280 NEXT I290 INPUT "Continue (Yes = 1, No = 0)”; I300 IF I ( )0 GOTO 210310 ENDNote that to compile the table of valuesof the solution for concrete values of on anda, we must know the limiting value C ofthis solution. The number C can be foundfrom the condition thatcx(1 —acosC) =sinC,which leads to the formulaC__Sin_1 ( Oi [1-a 1/1-q2.|.a2q,2])— 1—|—a2cz.2 °Here, if ex = 1, we have C = at/2-2 tan‘1a, but if cx < 1, then C z ot (1 — a).The limiting value C can be approximated(·‘i ’ by an exponential, namelyC — u z e‘V%,where ·y = oc (cos C -— a)/sin2 C. For smallcx’s the value of ·y is fairly great and is

128 Differential Equations in Applicationsapproximately given by the relationship/\JYN @(1-——a)But if ot :1, we have¤(1+¤2>Y: 1—a“ °For the step AX in the variation of theindependent variable X we must take anumber that does not exceed C. This require-ment is especially important for small oa’sand in the neighborhood of X = O. Belowwe give the protocol for calculating thevalues of the function u. = u (X).Solution of differential equation byRunge-Kutta second-order methodParameters:alpha = 1.0a = 0.3Step of independent variable = 0.5Initial values:of independent variable = 0of function = 0X u0 0.2 .1437181_4 .2299778.6 .2824244.8 .31468941 .3347111

Ch. 1. Construction of Differential Models 1291.2 .34720881.4 .35504031.6 .359961.8 .36305562 .36500542.2 .36623432.4 .36700912.600001 .36749782.800001 .3678063.000001 .36800053.200001 .36812323.400001 .36820063.600001 .36824943.800001 .3682802Continue (Yes = 1, No = O)? 1X u4.000001 .36829974.200001 .3683124.4 .36831974.6 .36832464.8 .36832765 .36832965.2 .36833085.399999 .36833165.599999 .36833215.799999 .36833245.999999 .36833266.199999 .36833276.399998 .36833286.599998 .36833286.799998 .36833296.999998 .36833299-0770

130 Differential Equations in Applications7.199998 .36833297.399997 .36833297.599997 .36833297.799997 .3683329Continue (Yes == 1, No = O)? 1gg u7.999997 .36833298.199997 .36833298.399997 .36833298.599997 .36833298.799996 .36833298.999996 .36833299.199996 .36833299.399996 .36833299.599996 .36833299.799996 .36833299.999995 .368332910.2 .368332910.4 .368332910.6 .368332910.8 .368332910.99999 .368332911 .19999 .368332911.39999 .368332911.59999 .36833291 1 . 79999 .3683329Continue (Yes = 1, No = O)? 0OKThe results of a numerical calculation ata = O.3 are presented graphically in Fig-ure 42. They show how an increase in X

Ch. 1. Construction of Differential Models 131influences the angle of lag of the loggingtruck. For clarity of exposition the scaleson the u and X axes are chosen to be differ-ent.Now let us determine the sweep of thelogging truck by using the concept of half-width of the road at the outer curb of a turn(this quantity was defined earlier as the al-gebraic sum OX — OA —}— W (Figure 41)).First we note thatOX2 = (-g—sin X—9»h cos 9)2—}—(£—c0sX-|—7tn sin6)2=h2 (2%-1-}.2--2-g;—-sin u) . -··This shows that the sweep decreases as Xgrows since the angle of lag, u, increases.Thus, the maximal halfwidth Bh of theroad can be determined from the followingformula for B:1 1 WB = I/’“2+w"tr+v7·In Figure 43 the solid curves reflect therelationship that exists between B-- W/hand oc for different values of Z., while thedashed curves show what must be thevalue of B -— L/h to guarantee the necessary"margin" at the inner curb of the turn. ForQi

132 Differential Equations in Applicationsu7 C`=CZ987¢?826CX = 76'=(Z56853Z9Of = Q 5a=g_; C=QO70/3720 50 XFig. 42W L/’ 7 #" F7 I /AEJ ;d°0IIA =2 / {cc-= Q3/ I’ 1/ ’ -// /,z’,/’/’/•/',#’0 ` 7 ’ Oc 0Fig. 43

Ch. 1. Construction of Differential Models 133every position of the logging truck on theroad we must have[5:.- %-(1—·cosu)-|—%’—1 L<—;(1—c0s C) +-5- (115)Next, since the value of C decreases with a,the angle of lag u proves to be the greatestwhen a, = 0, which corresponds to the simpl-est case of logging. Then (115) yields_. .... 2[1--2-<—;—(1-—cos C)|a=0-.:2——K6%2—,Now let us dwell on the results that followfrom the above line of reasoning. Here is atypical example that illustrates these re-sults.If a logging truck in which the distancebetween the front and back platforms is12 m follows a turn along an arc of a circleof radius 60 m, then oc = 0.2 and, in accor-dance with (109) N° is approximately 28°.If the width of the logging truck is 2.4 mand width of the load in the rear is 1.2 m,then for logs of approximately 24 m inlength (reckoning from the axis of thefront platform) we have X = 2, W/h =0.05, and L/h = 0.1. Thus, from Figure 43it follows that for all values of cz we haveB = 0.45 for the outer curb of the turn andB = 0.2 for the inner curb. Theoreticallythe necessary halfwidth of the road at the

134 Differential Equations in Applicationsouter curb of the turn is equal to 5.4 mand that at the inner curb to 2.4 m. If thelogging truck transports logs whose lengthis 14.4 m (reckoned from the axis of thefront platform) and whose bunch width inthe rear is 1.8 m, then in the case at handPt = 1.2. The value of [5 then proves to bethe same for the inner and outer curbs ofthe turn and equal to 0.22. Hence, as caneasily be seen, the necessary halfwidth ofthe road at the outer curb of the turn is2.64 m, while the same quantity at theinner curb, as in the previous case, is equalto 2.4 mi. This reasoning shows that thelonger the logs transported the wider theroad at a turn must be. For one, if we com-pare the two cases considered here, we seethat an increase in the length of the logs by9.6 m requires widening the road at a turnby 2.76 m so that the driver can drive thetruck along a curve whose length at the turnis approximately equal to the length of theroad’s center line. Practice has shown thatan inexperienced driver is not able todrive his truck along such a curve and needsa road whose total width at a turn is atleast 10.8 m (if the load transported is 24 mlong) for a truck width of 2.4 m.The theory developed above shows thatthe sweep of the logging truck is the great-est when the truck enters the turn, sincein this case the angle of lag grows. Thisconclusion also holds true in the situation

Ch. 1. Construction of Differential Models 135when the truck enters one section of a zigzagturn after completing the previous one atthe point of inflection. The results represent-ed graphically in Figure 43 correspond tothe case where prior to entering a turn thetractor unit and the trailer are positionedon a single straight line. But if there existsa nonzero initial angle of lag C0 caused bythe zigzag nature of the turn, we must selectthe initial condition in the differentialequation (114) in the form u (0) == ——C0.Then the required width of the road isdetermined from the following formula for B:rs: 1/ xw, $,-+2 gmc, --27.+ Q".We note that in the case of simple logtransportation, that is, a-:-0, it is impossibleto pass a turn with on greater than unity.However, for fairly large values of a thevalues on >1 become possible, they mustobey the following relationship: ot (1 ——a cos C) = sin C. Thus, the maximal valueof on is (1 — a2)**/2, with the practicalextremal value of ot being 1.25 at a == 0.5.We note in conclusion that for ot > 0.5considerable economy in the width of a roadis achieved by increasing a (see Figure 43).If the load is such that 2. is not much greaterthan unity, the value of oc is chosen suchthat the required halfwidth of the road atthe inner curb of any turn is always smallerthan at the outer curb.

Chapter 2Qualitative Methodsof Studying DifferentialModelsIn solving the problems discussed in Chap-ter 1 we constructed differential modelsand then sought answers by integrating theresulting differential equations. However,as noted in the Preface, the overwhelmingmajority of differential equations are notintegrable in closed (analytical) form.Hence, to study differential models of realphenomena and processes we need methodsthat will enable us to extract the necessaryinformation from the properties of thedifferential equation proper. Below withconcrete examples we show how in solvingpractical problems one can use the simplestapproaches and methods of the qualitativetheory of ordinary differential equations.2.1 Curves of Constant Directionof Magnetic NeedleLet us see how in qualitative integration,that is, the process of establishing the gen-eral nature of solutions to ordinary differen-tial equations, one can use a general proper-ty of such equations whose analogue, for

Ch. 2. Qualitative Methods 137example, is the property of the magneticfield existing at Earth’s surface. The readerwill recall that curves at Earth’s surfacecan be specified along which the directionof a magnetic needle is constant.Thus, let us consider the first-order or-dinary differential equation-3%-=f(¤¤, 11), (U6)where the function f (1:, y) is assumed single-valued and continuous over the set ofvariables sc and y within a certain domain Dof the (ec, y)-plane. To each point M (x, y)belonging to the domain D of functionf (x, y) the differential equation assignsa value of dy/dx, the slope K of the tangentto the integral curve at point M (:2:, y).Bearing this in mind, we say that at eachpoint M (:c, y) of D the differential equation(116) defines a direction or a line element.The collection of all line elements in D issaid to be the field of directions or theline element field. Graphically a linear ele-ment is depicted by a segment for whichpoint M (az, y) is an interior point and whichmakes an angle 9 with the positive directionof the x axis such that K = tan 9 =f (sc, y). From this it follows that geometri-cally the differential equation (116) expressesthe fact that the direction ofa tangent atevery point of an integral curve coincides withthe direction of the field at this point.

138 Differential Equations in ApplicationsTo construct the field of directions it hasproved expedient to use the concept of iso-clinals (derived from the Greek words for"equal" and "sloping"), that is, the set ofpoints in the (.2:, y)-plane at which thedirection of the field specified by the differ-ential equation (116) is the same.The isoclinals of the magnetic field atEarth’s surface are curves at each point ofwhich a magnetic needle points in the samedirection. As for the differential equation(116), its isoclinals are given by the equa-tionf (J:. y) = viwhere v is a varying real parameter.Knowing the isoclinals, we can approx-imately establish the behavior of theintegral curves of a given differential equa-tion. Let us consider, for example, thedifferential equation{11%-=w2—l-y·'*,which cannot be integrated in closed (ana-lytical) form. The form of this equationsuggests that the family of isoclinals is givenby the equationx2+y2=v, v>O,that is, the isoclinals are concentric circlesof radius I/v centered at the origin and ly-ing in the (as, y)-plne, At each point of

Ch. 2. Qualitative Methods 1399V gg? itFig. 44such an isoclinal the slope of the tangentto the integral curve that passes throughthis point is equal to the squared radius ofthe corresponding circle. This informationalone is sufficient to convey an idea of thebehavior of the integral curves of thegiven differential equation (Figure 44).We arrived at the final result quicklybecause the example was fairly simple.However, even with more complicated equa-tions knowing the isoclinals may prove tobe expedient in solvinga specific problem.Let us consider a geometric method ofintegration of differential equations of the

140 Differential Equations in Applicationstype (116). The method is based on usingthe geometric properties of the curvesgiven by the equationsf(¤¤· y)=0· (I0)0f(=v. y) 6f(==. 11) _‘—3;,—·+*—5‘,;·—*f(9?» ll)- 0- (L)Equation (I 0), that is, the "zero" isoclinalequation, specifies curves at whose pointsdy/dx = 0. This means that the points ofthese curves may prove to be points of maxi-ma or minima for the integral curves ofthe initial differential equation. This ox-plains why out of the entire set of isoclinalswe isolate the "zero" isoclinal.For greater precision in constructingintegral curves it is common to find theset of inflection points of these curves (pro-vided that such points exist). As is known,points of inflection should be sought amongthe points at which y" vanishes. EmployingEq. (116), we find that. _ 6f(¤=, y) 6f(¢» y) »y ·· ‘—5;‘+—w*y0 . 6* ,:;.%¥+Ji§.!%,·(_,,, y)_The curves specified by Eq. (L) are the pos-sible point—of-inflection curves.* Note, for* It is assumed here that integral curves that{ill a certain domain possess the property that only(que integral curve passes through each point of theomain.

Ch. 2. Qualitative Methods 141one, that a point of inflection of an integralcurve is a point at which the integral curvetouches an isoclinal.Thezcurves consisting of extremum points(maxima and minima points) and points ofinflection of integral curves break down thedomain of f into such subdomains S1,S2, ., Sm in which the first and secondderivatives of the solution to the differ-ential equation have definite signs. Ineach specific case these subdomains shouldbe found. This enables giving a roughpicture of the behavior of integral curves.As an example let us consider the differ-ential equation y’ = x + y. The equationof curve (I 0) in this case has the formas + y = 0, or y = -:1:. A direct checkverifies that curve (I 0) is not an integralcurve. As for curve (L), whose equation inthis case is y -l— x —}— 1 = 0, we find thatit is an integral curve and, hence, is nota point-0f—inflection curve.The straight lines (I 0) and (L) breakdown the (sc, y)·plane into three subdomains(Figure 45): S1 (y' >O, y" >O), to theright of the straight line (I 0), S 2 (y' < 0,y" > 0), between the straight lines (I0) and(L), and S3 (y' < 0, y" < 0), to the left ofthestraight line (L). The points ofminima of theintegral curves lie on the straight line (I 0).T 0 the right of (I 0) the integral curves pointupward, while to the left they point down-ward (left to right in Figure 45). There are

142 Differential Equations in Applications‘ SI0Y —¤S5 \ K-1Fig. 45no points of inflection. To the right of thestraight line (L) the curves are convex down-ward and to the left convex upward. Thebehavior of the integral curves on the wholeis shown in Figure 45.Note that in the given case the integralcurve (L) is a kind of "dividing" line, sinceit separates one family of integral curvesfrom another. Such a curve is commonlyknown as a separatrix.

Ch. 2. Qualitative Methods 1432.2 Why Must an Engineer KnowExistence and UniquenessTheorems?When speaking of isoclinals and point—of—inflection curves in Section 2.1, we tacitlyassumed that the differential equation inquestion had a solution. The problem ofwhen a solution exists and of when it isunique is solved by the so·called existenceand uniqueness theorems. These theoremsare important for both theory and practice.Existence and uniqueness theorems arehighly important because they guaranteethe legitimacy of using the qualitative meth-ods of the theory of differential equationsto solve problems that emerge in scienceand engineering. They serve as a basis forcreating new methods and theories. Oftentheir proof is constructive, that is, the meth-ods by which the theorems are proved sug-gest methods of finding approximate solu-tions with any degree of accuracy. Thus,existence and uniqueness theorems lie atthe base of not only the above-noted quali-tative theory of differential equations butalso the methods of numerical integration.Many methods of numerical solution ofdifferential equations have been developed,and although they have the common draw-back that each provides only a concretesolution, which narrows their practicalpotential, they are widely used. It must be

144 Differential Equations in Applicationsnoted, however, that before numericallyintegrating a differential equation one mustalways turn to existence and unique-ness theorems. This is essential to avoidmisunderstandings and incorrect conclu-sions.To illustrate what has been said, let ustake two simple examples,* but hrst let usformulate one variant of existence anduniqueness theorems.Existence theorem If in Eq. (116) func-tion f is defined and continuous on a boundeddomain D in the (x, y)—plane, then for everypoint (xo, yo) E D there exists a solution y (x)to the initial-value problem **dy — — 117E- — f (ac. y). y (wo.) —— yo. ( >that is defined on a certain interval con-taining point xo.Existence and uniqueness theorem Ifin Eq. (116) function f is defined and contin-uous on a bounded domain D in the (x, y)-* See C.E. Roberts, J r., "Why teach existence anduniqueness theorems in the first course of ordinarydifferential equations?", Int. J. Math. Educ. Sci.Technol. 7, No. 1: 41-44 (1976).** If we wish to find a solution of a differentialequation satisfying a certain initial condition (inour case the initial condition is y (xo) = yo), sucha problem issaid to be an initial-value problem.

Ch. 2. Qualitative Methods 145plane and satisfies in D the Lipschitz condi-tion in variable y, that is,liisv, yi)-—f(x, y1)|<L|y2-—y1l.with L a positive constant, then for everypoint (xo, yo) E D there exists a unique solu-tion y (x) to the initial-value problem (117)defined on a certain interval containingpoint xo.Extension theorem If the hypotheses ofthe existence theorem or the existence anduniqueness theorem are satisfied, then everysolution to Eq. (116) with the initial data(xo, y0) E D can be extended to a point thatlies as close to the boundary of D as desired.In the first case the extension is not necessarilyunique while in the second it is.Let us consider the following problem.Using the numerical Euler integrationmethod with the iteration scheme y,+1 =yi -}— hf (xi, yi) and step h = 0.1, solvethe initial-value problemy' == -—x/y, y (—— 1) = 0.21 (118)on the interval l-- 1, 3].Note that the problem involving theequation y' = —x/y emerges, for example,from the problem considered on p. 162 con-cerned with a conservative system consist-ing of an object oscillating horizontally in10-0770

146 Differential Equations in Applicationsa vacuum under forces exerted by linearsprings.To numerically integrate the initial·valueproblem (117) on the l·—-1, 2.8] intervalwe compile a program for building thegraph of the solution:10 REM Numerical integration of differentialequation20 DEF FNF(X, Y) :30 GOSUB 1110: REM Coordinate axes40 REM Next value of function1000 LINE —(FNX(X), FNY(Y)), 21010 IF X < 2.9 GOTO 1001020 LOCATE 23, 11030 END1100 REM Construction of coordinate axes1110 SCREEN 1, 1, 0: KEY OFF: CLS1120 DEF FNX(X) = 88 —l— 80*X1130 DEF FNY(Y) = 96 — 80*Y1140 REM Legends on coordinate axes1150 LOCATE 1, 11, 0 PRINT "Y"1160 LOCATE 3, 11: PRINT "1"1170 LOCATE 13, 39: PRINT "X"1180 LOCATE 23, 10: PRINT "-1"1190 FOR I = -1 TO 2: LOCATE 13, 10*1 +11: PRINT USING "4:t xt"; I;: NEXT I1200 REM Oy1210 DRAW "BM88, ONM -— 2, —{—8NM + 2,—|—8D16"1220 FORI = 1 TO 11: DRAW "NR2D16": NEXT1230 REM Ox

Ch. 2. Qualitative Methods 1471240 DRAW "BM319, 96NM -—— 7, ——2NM — 7—|—2L23"1250 FOR I = 1 TO 19: DRAW "NU2L16": NEXT1260 REM Input of initial data1270 LOCATE 25, 1: PRINT STRING$(40, "),1280 LOCATE 25, 1: INPUT, "xO =", X: INPUT;"dx =”, DX: INPUT; "yO =", Y1290 PSET (FNX(X), FNY(Y)), 21300 RETURNHere in line 20 the right—hand side of thedifferential equation is specified, and lines50 to 990 must hold the program of thenumerical integration of the differentialequation.In our case (the initial-value problem(118)) the beginning of the program hasthe following form:10 REM Euler’s method20 DEF FNF(X, Y) = —X/Y30 GOSUB 1100: REM Coordinate axes40 REM Next value of function100 Y = Y —|- DX*FNF(X, Y)110 X = X + DXThe results are presented graphically inFigure 46.Let us now turn to the existence theorem.For the initial-value problem (118), thefunction f (:2:, y) = —x/y is defined andcontinuous in the entire (x, y)—plane exclud-10*

H°°.N4 Q*<\`<oBO°§<¤ N <\¤ <¤5 " U U ¤· 5 TN · n9<¤SI ec<1·'\ E' In

Ch. 2. Qualitative Methods 149ing the :1: axis. Thus, in accordance withthe existence theorem, the initial-valueproblem (118) has a solution y (x) defined ona certain interval containing point xo =-1. This solution, according to the exten-sion theorem, can be extended to a valueof y (cc) close to the value y (x) =-· O. Asa result of numerical integration we havearrived at a solution of (118) defined on aninterval (a, b), with a< -1 and 1.3<b<1.4. However, allowing for the concreteform of the differential equation, we canspecify the true interval in which thesolution to the initial-value problem (118)exists. Indeed, since in the initial equationthe variables can be separated, we havey IS mm- —§ we0.21 -1Integrating, we get y = I/1.0441 —— 1:2.Hence, a solution to the initial—value prob-lem (118) exists only for | :2: | < I/1.0441,*:1 1.0218.Thus, by resorting to the existence theo-rem (and to the extension theorem) we wereable to "cut ofi" the segment on which thereis certain to be no solution of the initial-value problem. If we employ only numericalintegration, we arrive at an erroneousresult. The fact is that as the solutiony = y (as) approaches the sv axis, the angle

150 Differential Equations in Applicationsof slope of the curve tends to 90°. Therefore,in the time that the independent varia-ble x changes by 0.1, the value of y is able to"jump over" the as axis, and we find ourselveson an integral curve that differs from theoriginal. This happens because the Eulermethod allows for the angle of slope onlyat the running point.The following example is even moreinstructive. We wish to solve the initial-value problemy' = 3x?/yi y (-1) == --1 (M9)on the segment [— 1, 1]. The approach hereconsists in first employing the Euler methodand then an improved Euler method witha step h = 0.1 and an iteration schemeE/z+1 = Hz + hl (xi+1/21 yi+1/2)? withyi+1/2 = yi + hf ( ¤?z» yi)/2-We solve the initial-value problem (119)using the above program, whose beginningin the case at hand has the following form:10 REM Euler’s method20 DEF FNF(X, Y) =3·•=X·•=SGN(Y)·•·ABS(Y) A(1/3)30 GOSUB 1100: BEM Coordinate axes40 REM Next value of function100 Y = Y —l— DX*FNF(X, Y)110 X ¤= X —{— DXThe results are presented graphically inFigure 47.

Ch. 2. Qualitative Methods 151.16/*7Q6Q2*7 ·l26 ·Q2 0 QZ Q5 1 as-Q2*625·· IFig. 47As for the improved Euler method, thebeginning of the program has the followingform:10 REM Improved Euler’s method20 DEF FNF(X, Y) == 3*X*SGN(Y)*ABS(Y) A(1/3)30 GOSUB 1100: REM Coordinate axes40 REM Next value of function50 D2 = DX/2

152 Differential Equations in Applications91Q6622-7 -05 ·QZ0 az as 1 .7;‘QZ-06-7Fig. 48100 Y1 = Y + D2·•·FNF(X, Y)110 Y = Y -I— DX·¤=FNF(X —l— D2, Y1)120 X = X + DXThe results are presented graphically inFigure 48, and the diagram differs fromthat depicted in Figure 47.To look into the reason for such strikingdiscrepancy in the results, we integrate the

Ch. 2. Qualitative Methods 153initial-value problem. Separating the vari-ables, we getQ! xS 1]**/3 dn:-3S §d§,-1 -1or, finally, y = ;i;x". This result alreadysuggests that the solution via the Eulermethod gives the function yl (sc) == 1:3 whilethe solution via the improved Euler methodgives.2:3 if :1:<O,y2(x) Zl -.1:3 if ;1:>OBoth yl and yl are solutions to the initial-value problem (119), which means that thesolution of the initial—value problem con-sidered on the segment [-1, 1] is notunique.Let us now turn to the existence anduniqueness theorem in connection with thisproblem. First, we note that since the func-tion f (.2:, y) = 3;::%/y` is continuous in theentire (1:, y)-plane, the existence theoremimplies that the initial—value problem (119)has a solution defined on a segment con-taining point xl, = --1, and, according tothe extension theorem, this solution canbe extended to any segment. Further, since6f (ac, y)/0y = :cy‘2/3, the functionf (sv, y)=

154 Differential Equations in Applications3x:}/y satisfies the Lipschitz condition invariable y in any domain not containingthe 1: axis. If, however, a domain doescontain points belonging to the x axis, itis easy to show that the function doesnot satisfy the Lipschitz condition. Hence,from the existence and uniqueness theorem(and the extension theorem) it follows thatin this case the solution to the initial—valueproblem can be extended in a unique man-ner at least to the x axis. But since thestraight line y = O constitutes a singularintegral curve of the differential equationy' = Bx:}/y, we already know that as soonas y vanishes, there is no way in which wecan extend the solution to the initial-valueproblem (119) beyond point O(O, 0) ina unique manner.Thus, by resorting to the existence anduniqueness theorem (and the extensiontheorem) we were able to understand theresults of numerical integration, that is, ifwe are speaking of the uniqueness of thesolution of the initial—value problem (119)on the [-1, 1] segment, the solution existsand is defined only on the segment [-1, O].Generally, however, there can be severalsuch solutions.

Ch. 2. Qualitative Methods 1552.3 A Dynamical Interpretationof Second-Order DifferentialEquationsLet us consider the nonlinear differentialequationdzx dx1at:’(‘”»m‘) (*20)whose particular case is the second-order dif-ferential equation obtained on p. 72 whenwe considered pendulum clocks. We takea simple dynamical system consisting ofa particle of unit mass that moves along the:1: axis (Figure 49) and on which a forcef (sz, dx/dt) acts. Then the differentialequation (120) is the equation of motionof the particle. The values of zz: and dx/dtat each moment in time characterize thestate of the system and correspond to apoint in the (:1:, dx/dt)-plane (Figure 50),which is known as the plane of states orthe phase (az, dx/dt)—plane. The phase planedepicts the set of all possible states of thedynamical system considered. Each new stateof the system corresponds to a new point inthe phase plane. Thus, the changes in thestate of the system can be represented bythe motion of a certain point in the phaseplane. This point is called a representativepoint, the trajectory of the representativepoint is known as the phase trajectory,

156 Differential Equations in Applicationsda:f("’°>Ez/T)

O J;Fig. 49a'1 B-? (J;) gl?)-plane0 asFig. 50and the rate of motion of this point as thephase velocity.If we introduce the variable y = dx/dt,Eq. (120) can be reduced to a system oftwo differential equations:dgé- =y, =.f(¤>» y)· (121)If we take t as a parameter, then the solu-tion to system (121) consists of two func-tions, :1: (t) and y (t), that in the phase(:1:, y)-plane define a curve (a phase trajec-tory).

Ch. 2. Qualitative Methods 157It can be shown that system (121) andeven a more general system%f—=X(r¤» y)» {1]%-·=Y(¤¤» y)· (122)where the functions X and Y and theirpartial derivatives are continuous in a do-main D, posseses the property that if .1: (t)and y (t) constitute a'solution to the differ-ential system, we can writew=w(¢+C),y=y(¢+C). (123)where C is an arbitrary real constant, and(123) also constitute a solution to the samedifferential system. All solutions (123)with different values of C correspond to asingle phase trajectory in the phase (.1:, y)-plane. Further, if two phase trajectorieshave at least one common point, theycoincide. Here the increase or decrease inparameter t corresponds to a certain direc-tion of motion of the representative pointalong the trajectory. In other words, aphase trajectory is a directed, or oriented,curve. When we are interested in thedirection of the curve, we depict the direc-tion of the representative point along thetrajectory by placing a small arrow on thecurve.Systems of the (122) type belong to theclass of autonomous systems of differentialequations, that is, systems of ordinarydifferential equations whose right-hand

158 Differential Equations in Applicationssides do not explicitly depend on time t.But if at least in one of the equations ofthe system the right-hand side dependsexplicitly on time t, then such a system issaid to be nonautonomous.In connection with this classification ofdifferential equations the following remarkis in order. If a solution .2: (t) to Eq. (120)is a nonconstant periodic solution, thenthe phase trajectory of the representativepoint in the phase (x, y)—plane is a simpleclosed curve, that is, a closed curve withoutself—intersections. The converse is also true.If differential systems of the (122) typeare specified in the entire (x, y)—plane, then,generally speaking, phase trajectories willcompletely cover the phase plane withoutintersections. And if it so happens thatX (xm yo) ;'— Y ($01 y0) Z 0at a point Mo (xo, yo), the trajectory degen-erates into a point. Such points are calledsingular. In what follows we consider primar-ily only isolated singular points. A singu-lar point Mo (xo, yo) is said to be isolatedif there exists a neighborhood of this pointwhich contains no other singular points€X0€Pl> M0 (xm yo)-From the viewpoint of a physical inter-pretation of Eq. (120), the point Mo (xo, 0)is a singular point. At this point y =0 andf (xo, 0) = 0. Thus, in this case the isolatedsingular point corresponds to the state of

Ch. 2. Qualitative Methods 159a particle of unit mass in which both thespeed dx/dt and the acceleration dy/di =dzsc/dig of the particle are simultaneously ze-ro, which simply means that the particleis in the state of rest or in equilibrium.In view of this, singular points are alsocalled points of rest or points of equilibrium.The equilibrium states of a physical sys-tem constitute very special states of thesystem. Hence, a study of the types ofsingular points occupies an important placein the theory of differential equations.The first to consider in detail the clas-sification of singular points of differentialsystems of the (122) type was the distin-guished Russian scientist Nikolai E. Zhu-kovsky (1847-1921). In his master’s the-sis "The kinematics of a liquid body",presented in 1876, this problem emerged inconnection with the theory of velocitiesand accelerations of fluids. The modernnames of various types of singular points weresuggested by the great French mathemati-cian Jules H. Poincaré (1854-1912).Now let us try to answer the question ofthe physical meaning that can be attachedto phase trajectories and singular points ofdifferential systems of the (122) type. Forthe sake of clarity we introduce a two-dimensional vector field (Figure 51) definedby the functionV (rv. y) = X (wl y)i + Y (rv, y)i.

160 Differential Equations in Applications

P -1XYFig. 51where i and j are the unit vectors directedalong the x and y axes, respectively, ina Cartesian system of coordinates. At everypoint P (x, y) the field has two components,the horizontal X (x, y) and the verticalY (sc, y). Since dx/dt -—= X (x, y) and dy/dt:Y (as, y), the vector associated with eachnonsingular point P (x, y) is tangent atthis point to a phase trajectory.lf variable t is interpreted as time, vec-tor V can be thought of as the vector ofvelocity of a representative point movingalong a trajectory. Thus, we can assumethat the entire phase plane is filled withrepresentative points and that each phasetrajectory constitutes the trace of a movingrepresentative point. As a result we arriveat an analogy with the two-dimensionalmotion of an incompressible fluid. Here,since system (122) is autonomous, vector V

Ch. 2. Qualitative Methods 1.61at each fixed point P (sc, y) is time-inde-pendent and, therefore, the motion of thefluid is steady-state. The phase trajectoriesin this;case are simply the trajectories ofthe moving particles of fluid and the singu-lar points O, O', and O" (see Figure 51) arethose where the fluid is at rest.The most characteristic features of thefluid motion shown in Figure 51 are (1)the presence of singular points, (2) thedifferent patterns of phase trajectories nearsingular points, (3) the stability or instabil-ity at singular points (i.e. two possibili-ties may realize themselves: the particlesthat are in the vicinity of singular pointsremain there with the passage of time orthey leave the vicinity for other parts ofthe plane), and (4) the presence of closedtrajectories, which in the given case corre-spondmto periodic motion.These features constitute the main partof the phase portrait, or the completequalitative behavior pattern of Qthe phasetrajectories of a general—type system (122).Since, as noted earlier, differential equa-tions cannot generally be solved analytically,the aim of the qualitative theory of ordinarydifferential equations of the (122) type is tobuild a phase portrait as complete as possi-ble directly from the functions X (x, y)and Y (sc, y).11-0770

162. Differential Equations in Applications2.4 Conservative Systemsin MechanicsPractice gives us ample examples of thefact that any real dynamical systemdissipates energy. The dissipation usuallyoccurs as a result of some form of friction.But in some cases it is so slow that it can beneglected if the system is studied over a fair-ly small time interval. The law of energyconservation, namely, that the sum of kinet-ic and potential energies remains constant,can be assumed to hold true for such sys-tems. Systems of this kind are called con-servative. For example, rotating Earth maybe seen as a conservative system if we takea time interval of several centuries. But ifwe study Earth’s motion over several mil-lion years, we must allow for energy dissipa-tion related to tidal ilows of water in seasand oceans.A simple example of a conservative sys-tem is one consisting of an object movinghorizontally in a vacuum under forces ex-erted by two springs (Figure 52). lf as is thedisplacement of the object (mass m) fromthe state of equilibrium and the force withwhich the two springs act on the object (therestoring force) is proportional to :1:, theequation of motion has the formm§-Q-1+kx=o, 1i>o.

Ch. 2. Qualitative Methods 163rn \Fig. 52Springs of this type are known as linear,since the restoring force exerted by them isa linear function of x.If an object of mass m moves in a medi-um that exerts a drag on it (the dampingforce) proportional to the object’s velocity,the equation of motion for such a nonconser-vative system ism!£+¢i‘€+kx=—.0 c>o (124)d¢2 dz ’ ’Here we are dealing with linear damping,since the damping force is a linear functionof velocity dx/dt.If f and g are such arbitrary functions thatf(0) = O and g (O) = 0, the more generalequation,daz dxm‘·(W·+§(·&T)+f(x)~0» (125)can be interpreted as the equation ofmotion of an object of mass m undera restoring force ——j (x) and a dampingforce —g (dx/dt). Generally, these forces11*

164 Differential Equations in Applicationsare nonlinear; hence Eq. (125) can be con—sidered the basic equation of nonlinear me-chanics.Let us briefly examine the special case ofa nonlinear conservative system describedby the equationd2mg? +1* (w) =0» (126)where the damping force is zero and, hence,energy is not dissipated. From Eq. (126) wecan pass to the autonomous system3 __ gg _ _ f(¤¤)in "y’ at — m · (127)If we now exclude time t from Eqs. (127),we arrive at a differential equation for thetrajectory of the system in the phase plane:dy __ __ f (¤¤)—(E— my . (128)This equation can be written asmy dy = -j(x) dx. (129)Then, assuming that as = xo at t = to andy = yo, we can integrate Eq. (129) from toto t. The result is33i—nzz/3---1- m 2: ·-S f(E) dE2 • 2 'yf) - ...7xo

Ch. 2. Qualitative Methods 165which may be rewritten as1 1—§—my2+S f(§) d§=*g·”'»!/3+S 1‘(¤¤) dw-0 0(130)Note that my'!/2 := m(d:1:/dt)2/2 is the ki-netic energy of a dynamical system andxV(w) = S f(E)d§ (131)0is the system’s potential energy. Thus,Eq. (130) expresses the law of energy conser-vation:émz/2+V(¤¤) =E» (132)where E = my?/2 -}- V (.2:0) is the total ener-gy of the system. Clearly, Eq. (132) is theequation of the phase trajectories of sys-tem (127), since it is obtained by integratingEq. (128). Thus, different values of E corre-spond to diiierent curves of constant energyin the phase plane. The singular points ofsystem (127) are the points M`, (::1,,, 0),where :1:,, are the roots of the equation f (as) =0. As noted earlier, the singular points arepoints of equilibrium of the dynamical sys-tem described by Eq. (126). Equation (128)implies that the phase trajectories of thesystem intersect the ac axis at right angles,while the straight lines sc = 0:,, are parallel

166 Differential Equations in Applicationsto the as axis. In addition, Eq. (132) showsthat the phase trajectories are symmetricwith respect to the .2: axis.In this case, if we write Eq. (132) in theformy = 1]/% [E—V(¤v)l, (133)we can easily plot the phase trajectories.Indeed, let us introduce the (.2:, z)-plane,the plane of energy balance (Figure 53),with the z axis lying on the same verticalline as they axis of tl1e phase plane. We thenplot the graph of the function z = V (1:) andseveral straight lines z = E in the (x, z)-plane (one such straight line is depicted inFigure 53). We mark a value of E — V (:1:)on the graph. Then for a definite sc we multi-ply E — V (x) by 2/m and allow for formu-la (133). This enables us to mark the respec-tive values of y in the phase plane. Notethat since dx/dt = y, the positive directionalong any trajectory is determined by themotion of the representative point from leftto right above the az axis and from right toleft below the x axis.The above reasoning is fairly general andmakes it possible to investigate the equa-tion of motion of a pendulum in a mediumwithout drag, which has the form (see p. 72)—(E£+ksinx==0 (134)dia 3 ‘°

Ch. 2. Qualitative Methods 167Z z=V(z2)5-1/(1;) { FE11 11 101 ' "’{ I‘ I9 1 I1 I: 11 111/-$[1* - 1/(.:2] 10 .2:Fig. 53with k a positive constant.Since Eq. (134) constitutes a particularcase of Eq. (126), it can be interpreted asan equation describing the frictionless moetion in a straight line of an object of unitmass under a restoring force equal to

168 Differential Equations in Applications-—k sin :1:. In this case the autonomoussystem corresponding to Eq. (134) isd d .—a—?=y, —£—=—-ksinx. (135)The singular points are (O, O), (j rt, 0),(;|; Zn, O), . ., and the differential equa-tion of the phase trajectories of system (135)assumes the formdy __ _ k sin xYE" `—§_'Separating the variables and integrating,we arrive at an equation for the phase tra-jectories,-—%- yz-1+k (1 ·—-cos ac) = E.This equation is a particular case of Eq. (132)with m = 1, where the potential energydetermined by (131) is specified by therelationship3V(x) = S f(§)d§==k(1——cos:z:).0In the (x, z)—plane we plot the functionz = V (x) as well as several straight linesz = E (in Figure 54 only one such line,z = E = 2k, is shown). After determininga value of E —-— V (cv) we can draw a sketch

Ch. 2. Qualitative Methods 169zz=£_=2/{f £_Wx) Z==V{2:·)=k·—kc0sa:' 1{ 11 It II I-5.rrI -2zr wl _ m Zyr Jyrl as1 I I1 { I 11 I 3/ I I1 , I 1| 1‘ , 1 * © ~\J\J\ / 1Fig. 54of the trajectories in the phase plane byemploying the relationshipy=:I:l/2[E-—V(¤¤)I·

170 Difierential Equations in ApplicationsThe resulting phase portrait shows (seeFigure 54) that if the energy E varies from Oto 2k, the corresponding phase trajectoriesprove to be closed and Eq. (134) acquiresperiodic solutions. On the other hand, ifE >2lc, the respective phase trajectoriesare not closed and Eq. (134) has no periodicsolutions. Finally, the value E = 2lc corre-sponds to a phase trajectory in the phaseplane that separates two types of motion,that is, is a separatrix. The wavy lineslying outside the separatrices correspond torotations of a pendulum, while the closedtrajectories lying inside the regions bou_nd-ed by separatrices correspond to oscilla-tions of the pendulum.Figure 54 shows that in the vicinity ofthe singular points (j2rcm, O), m = 0, 1,2, . ., the behavior of the phase trajecto-ries —differs from that of the phase trajecto-ries in the vicinity of the singular points(—_l_—:rm, 0), where rz = 1, 2, . ..There are different types of singular points.With some we will get acquainted shortly.As for the above example, the singularpoints (5; 2:1tm, O), m = O, 1, 2, . . ., be-long to the vorteaypoint type, while thesingular points (jam, 0), rz. == 1, 2, . .,belong to the saddle-point type. A singularpoint of an autonomous differential systemof the (122) type is said to be a vortexpoint if there exists a neighborhood of thispoint completely filled with nonintersecting

Ch. 2. Qualitative Methods 171phase trajectories surrounding the point.A saddle point is a singular point adjoinedby a finite number of phase trajectories("whiskers") separating a neighborhood ofthe singular point into regions where thetrajectories behave like a family of hyper-bolas defined by the equation xy = const.Now let us establish the effect of linearfriction on the behavior of the phase tra-jectories of a conservative system. Theequation is%%—+c·—°gl§i—+ksinx=O, c>O.lf friction is low, that is, the pendulum isable to oscillate about the position ofequilibrium, it can be shown that the phasetrajectories are such as shown in Figure 55.But if friction is so high that oscillationsbecome impossible, the pattern of phasetrajectories resembles the one depicted inFigure 56.If we now compare the phase portrait ofa conservative system with the last two por-traits of nonconservative systems, we seethat saddle points have not changed (weconsider only small neighborhoods of singu-lar points), while in the neighborhood ofthe singular points (j2¤tm, O), m = O, 1,2, . ., the closed phase trajectories havetransformed into spirals (for low friction)or into trajectories that "enter" the singularpoints in certain directions (for high fric-

hgy5

l

174 Differential Equations in Applicationstion). In the first case (spirals) we havesingular points of the focal-point type andin the second, of the nodal-point type.A singular point of a two-dimensionalautonomous differential system of the gen-eral type (122) (if such a point exists)is said to be a focal point if there existsa neighborhood of this point that is com-pletely filled with nonintersecting phasetrajectories resembling spirals that "wind"onto the singular point either as t-» +<x>or as t-> -0o. A nodal point is a singularpoint in whose neighborhood each phasetrajectory behaves like a branch of a parabo-la or a half-line adjoining the point alonga certain direction.Note that if a conservative system hasa periodic solution, the solution cannot beisolated. More than that, if l` is a closedphase trajectory corresponding to a periodicsolution of the conservative system, thereexists a certain neighborhood of l` that iscompletely filled with closed phase tra-jectories.Note, in addition, that the above defi-nitions of types of singular points havea purely qualitative, descriptive nature.As for the analytical features of these types,there are no criteria, unfortunately, inthe general case of systems of the (122)types, but for some classes of differentialequations such criteria can be formulated.

Ch.· 2. Qualitative Methods 175A simple example is the linear systemi§%‘:a1$+b1!/» ‘g'%=’a2$+bzy•where al, bl, az, and bz are real constants.If the coefficient matrix of this sytem isnonsingular, that is, the determinantat b1`07Iaz bz lfthe origin O (O, O) of the phase plane isthe only singular point of the differentialsystem.Assuming the last inequality valid, we de-note the eigenvalues of the coefficient ma-trix by X1 and X2. It can then be demonstrat-ed that(1) if K1 and K2 are real and of the samesign, the singular point is a nodal point,(2) if X1 and K2 are real and of oppositesign, the singular point is a saddle point,(3) if X], and A2 are not real and are notpure imaginary, the singular point is a fo-cal point, and(4) if kl and K2 are pure imaginary, thesingular point is a vortex point.Note that the first three types of singularpoints belong to the so—called coarse singu-lar points, that is, singular points whosenature is not affected by small perturba-tions of the right—hand sides of the initialdifferential system. On the other hand,

176 Differential Equations in Applicationsa vortex point is a fine singular point; itsnature changes even under small perturba-tions of the right-hand sides.2.5 Stability of Equilibrium Pointsand of Periodic MotionAs we already know, singular points ofdifferent types are characterized by differentpatterns of the phase trajectories in sufficient-ly small neighborhoods of these points. Thereis also another characteristic, the stabilityof an equilibrium point, which providesadditional information on the behavior ofphase trajectories in the neighborhood ofsingular points. Consider the pendulum de-picted in Figure 57. Two states of equilib-rium are shown: (a) an object of mass m isin a state of equilibrium at the uppermostpoint, and (b) the same object is in a stateof equilibrium at its lowest point. Thefirst state is unstable and the second, stable.And this is why. If the object is in itsuppermost state of equilibrium, a slightpush is enough to start it moving with anever increasing speed away from the equilib-rium position and, hence, away from theinitial position. But if the object is in thelowest possible state, a push makes it moveaway from the position of equilibriumwith a decreasing speed, and the weaker thepush the smaller the distance by which the

Ch. 2. Qualitative Methods {77IIIIIIIQ;Fig. 57object is displaced from the initial posi-tion.The state of equilibrium of a physicalsystem corresponds to a singular point inthe phase plane. Small perturbations at anunstable point of equilibrium lead to largedisplacements from this point, while ata stable point of equilibrium small pertur-bations lead to small displacements. Start-ing from these pictorial ideas, let usconsider an isolated singular point of sys-tem (122), assuming for the sake of simplici-ty that the point is at the origin O (O, O)of the phase plane. We will say that thissingular point is stable if for every positive Rthere exists a positive rg}? such that12-0770

180 Differential Equations in ApplicationsNikolai G. Chetaev (1902-1959) wrote: *...If a passenger plane is being designed, acertain degree of stability must be providedfor in the future movements of the plane sothat it will be stable in {light and accident-free during take-off and landing. The crank-shaft must be so designed that it does notbreak from the vibrations that appear in realconditions of motor operation. To ensure thatan artillery gun has the highest possible ac-curacy of aim and the smallest possible spread,the gun and the projectiles must be con-structed in such a manner that the trajectoriesof projectile flight are stable and the projec-tiles fly correctly.Numerous examples can be added to thislist, and all will prove that real movementsrequire selecting out of the possible solutionsof the equations of motion only those thatcorrespond to stable states. Moreover, if wewish to avoid a certain solution, it is advisableto change the design of the correspondingdevice in such a way that the state of motioncorresponding to this solution becomes un-stable.Returning to the pendulum depicted inFigure 57, we note the following curiousand somewhat unexpected fact. Researchhas shown that the upper (unstable) posi-tion of equilibrium can be made stable by* See N.G. Chetaev, Stability of Motion (Moscow:Nauka, 1965: pp. 8-9 (in Russian).

Ch. 2. Qualitative Methods 181introducing vertical oscillations of thepoint of suspension. More than that,not only the upper (vertical) position ofequilibrium can be made stable but also anyother of the pendulum’s positions (for one,a horizontal position) by properly vibratingthe point of suspension. *Now let us turn to a concept no less im-portant than the stability of an equilibriumpoint, the concept of stability of periodicmovements (solutions). Let us assume thatwe are studying a conservative system thathas periodic solutions. In the phase planethese solutions are represented by closedtrajectories that completely fill a certainregion. Thus, to each periodic motion ofa conservative system there correspondsa motion of the representative point alonga fixed closed trajectory in the phase plane.Generally, the period of traversal ofdifferent trajectories by representativepoints is different. In other words, the periodof oscillations in a conservative systemdepends on the initial data. Geometricallythis means that two closely spaced repre-sentative points that begin moving ata certain moment t = to (say, at the x axis)will move apart to a certain finite distance* The reader can find many cxancplts cf stabili-zation of different 15 pcs of pcndulums in the bookby T.G. Strizhak, Methods of Investigating Dynam-ical Systems of the “Pendulum” Type (Alma-Ata:Nauka, 1981) (in Russian).

178 Differential Equations in Applications.9VWI W`JFig. 58every phase trajectory originating at theinitial moment t = to at a point P lyinginside the circle :1:2 —i- y2 = r2 will lieinside the circle :::2 + y2 = R2 for all t > to(Figure 58). Without adhering to rigorousreasoning we can say that a singular pointis stable if all phase trajectories that arenear the point initially remain there withthe passage of time. Also, a singular pointis said to be asymptotically stable if it isstable and if there exists a circle x2 —I—y2 = rf, such that each trajectory that attime t = to lies inside the circle convergesto the origin as t-> —{—oo. Finally, ifa singular point is not stable, it is said tobe unstable.

Ch. 2. Qualitative Methods 179A singular point of the vortex type isalways stable (but not asymptotically).A saddle point is always unstable. InFigure 55, which illustrates the behaviorof the phase trajectories for pendulum oscil-lations in a medium with low drag, the sin-gular points, focal points, are asymptotical-ly stable; in Figure 56 the singular points,which are nodal points, are also asymptoti-cally stable.The introduced concept of stability of anequilibrium point is purely qualitative,since no mention of properties referring tothe behavior of phase trajectories has beenmade. As for the concept of asymptoticstability, if compared with the notion ofsimple stability, it is additionally neces-sary that every phase trajectory tend tothe origin with the passage of time. How-ever, in this case, too, no conditions areimposed on how the phase trajectory mustapproach point O (O, O).The concepts of stability and asymptoticstability play an important role in appli-cations. The fact is that if a device isdesigned without due regard for stabilityconsiderations, when built it will be sensi-tive to the very smallest external perturba-tions, which in the iinal analysis may leadto extremely unpleasant consequences. Em-phasizing the importance of the concept ofstability, the well-known Soviet specialistin the field of mathematics and mechanics12•

182 Differential Equations in Applicationsy 8(:*2P5 8Y} gl·`ig. 59with the passage of time. But it also mayhappen that these points do not separate.To distinguish between these two possibil-ities, the concept of stability in the senseof Lyapunov is introduced for periodicsolutions. The essence of this concept liesin the following. If knowing an a-neighbor-hood (with a as small as desired) of a pointM moving along a closed trajectory I`(Figure 59) * ensures that we: know a mov-ing 6 (s)-neighborhood of the same point Msuch that every representative point thatinitially lies in the 6 (s)-neighborhood willnever leave the e-neighborhood with thepassage of time, then the periodic solution* An s—neighborhood of a point M is understoodto be a disk of radius 6 centered at point M,

Ch. 2. Qualitative Methods 183corresponding to l` is said to be stable inthe sense of Lyapunov. If a periodic solutionis not stable in the sense of Lyapunov, it issaid to be unstable in the sense of Lyapunov.When it comes to periodic solutions thatare unstable in the sense of Lyapunov, wemust bear in mind that they still possesssome sort of stability, orbital stability,which means that under small variations ofinitial data the representative point trans-fers from one phase trajectory to anotherlying as close as desired to the initiallyconsidered trajectory.Examples of periodic solutions that arestable in the sense of Lyapunov are thosethat emerge, for instance, when we considerthe differential equation that describes thehorizontal movements of an object of massni in a vacuum with two linear springs actingon the object (Figure 52). An example ofperiodic solutions that are unstable in thesense of Lyapunov but are orbitally stableis the solutions of the differential equa-tion (134), which describes the motion ofa circular pendulum in a medium withoutdrag.In the first case the oscillation perioddoes not depend on the initial data and isfound by using the formula T = 2nl/m/lc.In the second the oscillation period dependson the initial data and is expressed, as weknow, in terms of an elliptic integral of thefnrst kind taken from 0 to at/2.

184 Differential Equations in ApplicationsFinally, we note that the question ofwhether periodic movements are stable inthe sense of Lyapunov is directly linkedto the question of isochronous vibrations. *2.6 Lyapunov FunctionsIntuitively it is clear that if the totalenergy of a physical system is at its mini-mum at a point of equilibrium, the pointis one of stable equilibrium. This idea liesat the base of one of two methods used instudying stability problems, both suggestedby the famous Russian mathematician Alek-sandr M. Lyapunov (1857-1918). This meth-od is known as Lyapunov’s direct, orsecond, method for stability investigations. **We illustrate Lyapunov’s direct methodusing the (122) type of system when theorigin is a singular point.Suppose that l` is a phase trajectory ofsystem (122). We consider a function V =V (x, y) that is continuous together with itsfirst partial derivatives 0V/6x and 6V/6y* See, for example, the book by V. V. Amel’kin,N. A. Lukashevich, and A.P. Sadovskii, Non-linear Vibrations in Second-Order Systems (Minsk:Belorussian Univ. Press, 1982) (in Russian).** The reader will End many interesting examplesof stability investigations involving differentialmodels in the book by N. Rouche, P. Habets, andM. Laloy, Stability Theory by Lyapunov`s DirectMethod (New York: Springer, 1977).

Ch. 2. Qualitative Methods 185in a domain containing I` in the phase plane.If the representative point (sc (t), y (1:))moves along curve I`, then on this curvethe function V (x, y) may be considereda function of t, with the result that therate at which V (:1:, y) varies along l` isgiven by the formuladV GV dx OV dy _ OV·g·;=‘g jg‘l"‘5,;· §—·gX($» H)where X (.2:, y) and Y (sc, y) are the right-hand sides of system (122).Formula (136) is essential in the realiza-tion of Lyapunov’s direct method. Thefollowing concepts are important for thepractical application of the method.Suppose that V = V(x, y) is continu-ous together with its first partial deriva-tives 6V/Ox and 0V/dy in a domain G con-taining the origin in the phase plane, withV (0, O) = O. This function is said to bepositive (negative) definite if at all pointsof G except the origin V (sc, y) it is positive(negative). But if at points of G we haveV (sc, y)/> 0 (Q O), the function V = V (1:,y) is said to be nonnegative (nonpositive).For example, the function V defined by theformula V (sc, y) = x2 + y2 and consideredin the (az, y)-plane is positive definite, whilethe function V (as, y) = $2 is nonnegativesince it vanishes on the entire y axis.

186 Differential Equations in ApplicationsIf V (x, y) is positive definite, we canrequire that at all points of the domainG\O the following inequality hold true:OV 2 0V 2(H) HE) * °·This means that the equation z = V (x, y)can be interpreted as the equation of a sur-face resembling a paraboloid that touchesthe (x,y)-plane at point O (0,0) (Figure60). Generally, the equation z = V (ay, y)with a positive definite V may specifya surface of a more complex structure. Onesuch surface is shown in Figure 61, wherethe section of the surface with the planez = C results not in a curve but in a ring.lf a positive definite function V (0:, y) issuch thatWo, y>=2—’-§,%x<x, y>+i‘i§-ii Y as y> weis nonpositive, V is said to be the Lyapunovfunction of system (122). We note here thatin view of (136) the requirement that W benonpositive means that dV/dtg 0 and,hence, the function V = V (zz, y) does notincrease along the trajectory I` in theneighborhood of the origin.Here is a result arrived at by Lyapunov:if for system (122) there exists a Lyapunovfunction V (x, y), then the origin, which is

Ch. 2. Qualitative Methods {87zcv 0 5/Fig. 60z .WlI\\& yFig. 61a singular point, is stable. If the positivedefinite function V = V (sc, y) is such thatfunction Wdefined via (137) is negative defi-nite, then the origin is asymptotically stable.We will shew with an example how to

188 DiHerential Equations in Applicationsapply the above result. Let us consider theequation of motion of an object of unitmass under a force exerted by a spring,which in view of (124) can be written in theform-(3;;;+clg-?+kx=0, c>0. (138)The reader will recall that in this equationc > 0 characterizes the drag of the mediumin which the object moves and k > 0characterizes the properties of the spring(the spring constant). The autonomous sys-tem corresponding to Eq. (138) has theform-3-?=y, $4:-—kx-cy. (139)For this system the origin of the phase(x, y)-plane is the only singular point. Thekinetic energy of the object (of unit mass)is y2/2 and the potential energy (i.e. theenergy stored by the spring) isS kg dg Z TQ- azz.0This implies that the total energy of thesystem isVMS, y)=% y2+é— M2- (140)It is easy to See that V specified by (140)

Ch. 2. Qualitative Methods 189is a positive definite function. And sincein the given case0-g-§X(¢» y>+;,g-Y<¤¤» y)=/~¤¤¤y+y(—k¤>-vy>= —cy2<0.this function is the Lyapunov function ofsystem (139), which means that the singu-lar point O (O, O) is stable.In the above example the result wasobtained fairly quickly. This is not alwaysthe case, however. The formulated Lyapu-nov criterion is purely qualitative andthis does not provide a procedure forEnding the Lyapunov function even if weknow that such a function exists. Thismakes it much more difficult to determinewhether a concrete system is stable or not.The reader must bear in mind that theabove criterion of Lyapunov must be seenas a device for finding effective indicationsof equilibrium. Many studies have beendevoted to this problem and a number ofinteresting results have been obtained inrecent yea1·s.*2.7 Simple States of EquilibriumThe dynamical interpretation of second-order differential equations already implies" The interested reader can refer to the book byE.A. Barbashin, Lyapunov Functions (Moscow:Nauka, 1970) (in Russian).

190 Differential Equations in Applicationsthat investigation of the nature of equilibri-um states or, which is the same, the singularpoints provides a key for establishing thebehavior of integral curves.It is also clear that, generally speaking,differential equations cannot be integratedin closed form. We need criteria that willenable us to determine the type of a sin-gular point from the form of the initialdifferential equation. Unfortunately, asa rule it is extremely difficult to {ind suchcriteria, but it is possible to isolate certainclasses of differential equations for whichthis can be done fairly easily. Below weshow, using the example of an object of unitmass subjected to the action of linearsprings and moving in a medium withlinear drag, how some results of the quali-tative theory of differential equations canbe used to this end. But first let us discussa system of the (122) type. It so happensthat the simplest case in establishing thetype of a singular point is when the Jacobianor functional determinant6X 6XT9? 75]J ("’* y): ay ar"`6T T9?is nonzero at the point.If (x* , y*) is a singular point of system (122)and if J* = J (x*, y*) ak 0, then the typeof the singular point, which in the given

Ch. 2. Qualitative Methods 191case is called a simple singular point,depends largely on the sign of constant J *.For instance, if J * is negative, the singularpoint (x*, y*) is a saddle point, and if .I`* ispositive, the singular point may be a vor-tex point, a nodal point, or a focal point.The singular point may be a vortex pointonly if the divergence6X GYD(w, y) =q;+—g,-7vanishes at the singular point, that is, onlyif D* = D (;z:*, y*) = O. Note, however,that the condition D* = 0 is generallyinsufficient for the singular point (x*, y*) tobe a vortex point. For a vortex point tobe present certain additional conditionsmust be met, conditions that include high-er partial derivatives. And, generally,there can be an infinite number of suchconditions. But if functions X and Y arelinear in variables as and y, the conditionD* = 0 becomes sufficient for the singularpoint (.2:*, y*) to be a vortex point.If J * > 0 but the singular point (42:*, y*)is not a vortex point, a sufficiently smallneighborhood of this point is filled withtrajectories that either spiral into this pointor converge to it in certain directions. Here,if D* > 0, the singular point is reached asi—> --o¤ and is unstable while if D* < 0,the singular point is reached as i-> +¤oand proves to be stable. If the phase tra-

192 Differential Equations in Applicationsjectories that reach the singular point arespirals, we are dealing with a focal point,but if the integral curves converge to a sin-gular point along a certain tangent, thepoint is a nodal point (Figure 62).Irrespective of the sign of Jacobian .}*,the tangents to the trajectories of the differ-ential system (122) at a singular point (x*,y*) can be found from what is known as thecharacteristic equation0Y(=¤*. y*) ~ 6Y(¤>*. y*)~tg; Bx x + Oy y—— 141)Q; 6X (=¤*» y*) ~ 6X (=¤*» y*) ~ ° (0:1: w+ 6y ywhere,.;-:;::-.10*, ;7=y—y*. (142)If X and Y contain linear terms, thepartial derivatives in Eq. (141) act as co-efficients of x and y in the system obtainedfrom system (122) after introducing thesubstitution (142).Equation (141) is homogeneous. Hence,if we introduce the slope 2. =.· of what isknown as exceptional directions, we havethe following quadratic equation for find-ing K:XQK2 + (X; —— Y;) A. — Y; = O. (143)The discriminant of this equation isA = (X; —l— Y;)2 — 4.}* == D*2 — 4.}*.

QIk%— HgII¤» R °'° $-3CQ __ * ··xI QS"II ·~>°`°I¤» *35 ¤Qva __ IL]I QIII ~m¤¤¤» ¤~ ° .2%Ln13-0770

194 Differential lriquations in ApplicationsHence, if J* <O, which corresponds toa saddle point, Eq. (143) always gives tworeal exceptional directions. But if J * > 0,there are no real exceptional directions,or there are two, or there is one 2—folddirection. In the iirst case the singularpoint is either a vortex point or a focal point.The existence of real exceptional direc-tions means (provided that J * is positive)that there is a singular point of the nodaltype. For one thing, if there are two realexceptional directions, it can be provedthat there are exactly two trajectories (oneon each side) whose tangent at the singularpoint is one of the exceptional straight lines(directions) while all the other trajecto-ries "enter" the singular point touching theother exceptional straight line (Figure 62a).If A = O and Eq. (143) is not an identity,we have only one exceptional straight line.The pattern of the trajectories for this caseis illustrated by Figure 62b. It can beobtained from the previous case when thetwo exceptional directions coincide. Thesingular point divides the exceptionalstraight line into two half—lines, ll and Z2,while the neighborhood of the singular pointis divided into two sectors, one of which iscompletely filled with trajectories that"enter" the singular point and touch ll andthe other is completely filled with trajecto-ries that "enter" the singular point andtouch Z2. The boundary between the sectors

Ch. 2. Qualitative Methods 195consists of two trajectories, one of whichtouches I1 at the singular point and the othertouches L2 at this point.lf in Eq. (143) all coeflicients vanish, wearrive at an identity, and then all thestraight lines passing through the singularpoint are exceptional and there are exactlytwo trajectories (one on each side) thattouch each of these straight lines at thesingular point. This point (Figure 62c) issimilar to the point with one 2-fold realexceptional direction.2.8 Motion of a Unit—Mass ObjectUnder the Action of Linear Springsin a Medium with Linear DragAs demonstrated earlier, the differentialequation describing the motion of a unit-mass object under the action of linearsprings in a medium with linear drag hasthe form2.‘}E;i+c%§+kx=o. (144)So as not to restrict the differential model(144) to particular cases we will not tix thedirections in which the forces ——c (dx/dt)and —-kx act. As shown earlier, withEq. (144) we can associate an autonomoussystem of the form-£§—=y, -%=·—kx—cy. (145)13*

196 Differential Equations in ApplicationsIf we now exclude the trivial case withlc = O, which we assume is true for themeantime, the differential system (145) hasan isolated singular point at the origin.System (145) is a particular case of thegeneral system (122). In our concrete exam-plethe Jacobian J (:1:, y) = lc, and the divergenceD (xr, y) : ——c. The characteristic equa-tion assumes the formX2-{-cli.-|-k=0,where A =· cg — 4k is the discriminant ofthis equation. In accordance with the resultsobtained in Section 2.7 we arrive at thefollowing cases.(1) lf lc is negative, the singular point isa saddle point with one positive and onenegative exceptional direction. The phasetrajectory pattern is illustrated in Fig-ure 63, where we can distinguish betweenthree different types of motion. When theinitial conditions correspond to point a,at which the velocity vector is directed tothe origin and the velocity is sufficientlygreat, the representative point moves alonga trajectory toward the singular point ata decreasing speed; after passing the originthe representative point moves away fromit at an increasing speed. If the initialvelocity decreases to a critical value, which

Ch. 2. Qualitative Methods 197y=.i:0E` ·¤` CaFig. 63corresponds to point b, the representativepoint approaches the singular point ata decreasing speed and "reaches" the originin an infinitely long time interval. Finally,if the initial velocity is lower than thecritical value and corresponds, say, to pointC, the representative point approaches theorigin at a decreasing speed, which vanishesat a certain distance xl from the origin.At point (xl, O) the velocity vector reversesits direction and the representative pointmoves away from the origin.If the phase point corresponding to theinitial state of the dynamical system liesin either one of the other three quadrants,the interpretation of the motion is obvious.

198 Differential Equations in Applicationsyar:xFig. 64fjFig. 65(2) If lc > O, then J * is positive and thetype of singular point depends on thevalue of c. This leaves us with the fol-lowing possibilities:(2a) lf c = O, that is, drag is nil, thesingular point is a vortex point (Figure 64).The movements are periodic and theiramplitude depends on the initial condi-tions.

Ch. 2. Qualitative Methods 199(2b) If c > O, that is, damping is posi-tive, the divergence D : OX/0.2; + GY/fiy isnegative and, hence, the representativepoint moves along a trajectory toward theorigi11 and reaches it in an infinitely longtime interval.More precisely:(2b,) If A <O, that is, cg < 4k, the sin--gular point proves to be a focal point(Figure 65) and, hence, the dynamicalsystem performs damped oscillations aboutthe state of equilibrium with a decayingamplitude.(2b2) If A :0, that is, c2 :4k, thesingular point is a nodal point with a singlenegative exceptional direction (Figure 66).The motion in this case is aperiodic and cor-responds to the so—called critical damping.(2b,,) If A > O, that is, cg > 4k, thesingular point is a nodal point with twonegative exceptional directions (Figure 67).Qualitatively the motion of the dynamicalsystem is the same as in the previous caseand corresponds to damped oscillations.From the above results it follows thatwhen c >O and k>O, that is, drag is pos-itive and the restoring force is attractive,the dynamical system tends to a state ofequilibrium and its motion is stable.(2c) lf c < O, that is, damping is nega-tive, the qualitative pattern of the phasetrajectories is the same as in the case (2b),the only difference being that here the

200 Differential Equations in Applicationsy=.z‘0 »Z'Fig. 66_;/=.i~.... \` I _Z.Fig. 67

Ch. 2. Qualitative Methods 201k .1*.-0*V %Eg #*~Fig. 63y=.2§` i LvC > OFig. 69dynamical system ceases to be stable.Figure 68 contains all the above resultsand the dependence of the type of singular

202 Diflerential Equations in Applicationspoint on the values of parameters c and k.Note that the diagrams can also be inter-preted as a summary of the results of studiesof the types of singular points of system (122)when J* #0 at c = —D* and lc = J*.However, the fact that c = O does notgenerally mean that system (122) possessesa vortex point, and the fact that k = 0 doesnot mean that the system of a general typehas no singular point. These cases belongto those of complex singular points, whichwe consider below.Returning to the dynamical system con-sidered in this section, we note that if k = 0(c qé O), the autonomous system (145) cor-responding to Eq. (144) assumes the formdx dyW z ya `(T? = ""cy•This implies that the straight line y = 0 isdensely populated by singular points, withthe phase trajectory pattern shown inFigure 69.Finally, if k = c = 0, Eq. (144) assumesthe formdzxaz? *-0-The respective autonomous system isda: dy

Ch. 2. Qualitative Methods 203Here, as in the previous case, the .1: axis isdensely populated by singular points. Therespective phase trajectory pattern isshown in Figure 70.2.9 Adiabatic Flowof a Perfect Gas Through a Noazleof Varying Cross SectionA study of the flow of compressible viscousmedia is highly important from the practi-cal viewpoint. For one thing, such flowemerges in the vicinity of a wing and fuse-lage of an airplane; it also influences theoperation of steam and gas turbines, jetengines, the nuclear reactors.Below we discuss the flow of. a perfectgas through a nozzle with a varying crosssection (Figure 7-1); the specific heat capaci-ty of the gas is cp and the nozzle’s— varyingcross-sectional area is denoted by A. Theflow is interpreted as one-dimensional, thatis, all its properties are assumed to be uni-form in a single cross section of the nozzle.Friction in the boundary layer is causedby the tangential stress 1: given by the for-mulat = qpvz/2, (146)where q is the friction coefficient depend_ingbasically on the Reynolds number butassumed constant along the nozzle, p the

204 Differential Equations in Applications_;/-.5*.z·Fig. 70as

/Fig. 71flux density, and v the flow velocity. Final-ly, we assume that adiabaticity conditionsare met, that is, drag, combustion, chemi-cal transformations, evaporation, and con-densation are excluded.One of the basic equations describingthis type of flow is the well-known conti-

Ch. 2. Qualitative Methods 205nuity equation, which in the given cseis written in the formw = pA v, (147)where the flux variation rate w is assumedconstant. From this equation it followsthatdp dA dv _7+7-}--7-0. (148)Let us now turn to the equation for theenergy of steady—state flow. We note thatgenerally such an equation links the exter-nal work done on the system and the actionof external heat sources with the increasein enthalpy (heat content) in the flow andthe kinetic and potential energies. In ourcase the flow is adiabatic; hence, the energy-balance equation can be written in the form0=w(h-l—dh)—-—wh—{— w [v2/2 + d (122/2)] ——- wvz/2,ordh —i— d (02/2) : O, (149)where h is the enthalpy of the flow (thethermodynamic potential) at absolute tem-perature T. But in Eq. (149) dh = c,,dTand, therefore, we can write the equationfor the flow energy asc,,dT + d (v2/2) = O. (150)

206 Differential Equations in ApplicationsNow let us derive the momentum equa-tion for the flow. Note that here the commonapproach to problems involving a steady-state flow is to use Newton’s second law.Assuming that the divergence angle of thenozzle wall is small, we can write themomentum equation in the formpA+pdA—(p+dp)(A-l—dA)—1:dA= w dv,or—A dp —-— dA dp — 1 dA = w dv, (151)where p is the static pressure.The term dA dp in Eq. (151) is of ahigher order than the other terms and,therefore, we can always assume that themomentum equation for the flow has theform-—A dp -—- 1: dA = w dv. (152)If we denote by D the hydraulic diameter,we note that its variation along the nozzle’saxis is determined by a function F such thatD = F (:1:), where x is the coordinate alongthe nozzle’s axis. From the definition of thehydraulic diameter it follows thatdA 4da:-1-I-zi . (153)Noting that pvz/2 = ·ypM2, where Y is thespecific heat ratio of the medium, and M

Ch. 2. Qualitative Methods 207is the Mach number, we can write formula(146) thus:1 := qypM2. (154)Combining this with Eqs. (147) and (153),we arrive at the following represen—tation for the momentum equation (152):dp da: dv? __-1;+ YM-'*(4q—j—-|——;;)-O. (155)Denoting the square of the Mach number byy, employing Eqs. (148), (150), and (155),and performing the necessary algebraictransformations, we arrive at the followingdifferential equation:..1 I@(1+-L——y) (wry-F (=•¤>>-‘k’-:-—-£-..-.-.- (156)dx (i-·y) F (¤¤) ’where the prime stands for derivative ofthe respective quantity.*The denominator of the right-hand side ofEq. (156) vanishes at y = 1, that is, whenthe Mach number becomes equal to unity.This means that the integral curves of thelast equation intersect what is known as thesonic line and have vertical tangents at theintersection points. Since the right-hand* See J. Kestin and S.K. Zaremba, "One-dimen-sional high—speed flows. Flow patterns derivedfor the flow of gases through nozzles, includingcompressibility and viscosity elfects,” AircraftEngm. 25, No. 292: 172-175, 179 (1953).

208 Differential Equations in Applicationsside of Eq. (156) reverses its sign in theprocess of intersection, the integral curves“ilip over" and the possibility of inflectionpoints is excluded. The physical meaningof this phenomenon implies that alongintegral curves the value of a: must increasecontinuously. Hence, the section on whichthe integral curves intersect the sonic linewith vertical tangents must be the exitsection of the nozzle. Thus, the transitionfrom subsonic flow to supersonic (and back)can occur inside the nozzle only througha singular point with real exceptional di-rections, that is, through a saddle point ora nodal point.The coordinates of the singular points ofEq. (156) are specified by the equationsy* :1. F’(¤¤*> = v<1»which imply that these points are situatedin the divergiug part of the nozzle. A saddlepoint appears if J* is negative, that is,F”(:1:*) > O. Since q is a sufficiently smallconstant, a saddle point appears nea1· thethroat of the nozzle. A nodal point, on theother hand, appears only if F" (x*) is neg-ative. Thus, a nodal point emerges in thepart of the nozzle that lies behind an in-flection point of the nozzle’s profile or, inpractical terms, at a certain distance fromthe throat of the nozzle, provided that theproiile contains an inflection point.

Ch. 2. Qualitative Methods 209From the characteristic equationF(¤¢*) K2 + 2qv (V +1)%—-2(Y +i)F”(¢*) =0we can see that the slopes of the two excep-tional directions are opposite in sign in thecase of a saddle point and have the samesign (are negative) in the case of a nodalpoint. This means that only a saddle pointallows for a transition from supersonic tosubsonic velocities and from subsonic tosupersonic velocities (Figure 72). The caseof a nodal point (Figure 73) allows for acontinuous transition only from supersonicto subsonic flow.Since Eq. (156) cannot be integrated inclosed form, we must employ numericalmethods of integration in any further dis-cussion. It is advisable in this connectionto begin the construction of the four sepa-ratrices of a saddle point as integral curvesby allowing for the fact that the singularpoint itself is a point, so to say, from whichthese integral curves emerge. Such a con-struction is indeed possible since the char-acteristic equation provides us with thedirection of the two tangents at the singularpoint S (:1:*, l). lf this is ignored and themotion monitored as beginning at pointsa and b in Figure 72, which lie on differentsides of a separatrix, the correspondingpoints move along curves on and [5 thatstrongly diverge and, hence, provide no in-14-0770

210 Differential Equations in Applications:;~1Z F Z22)ty Ly:.-F/cv/7 I S(2I/ / Ib { I /3/ I{Ct .Z‘* ,9Fig. 72!=”z.S’7IIII0 1** .2*Fig. 73formation about the integral curve (theseparatrix) that "enters" point S. On the

Ch. 2. Qualitative Methods 211other hand, if we move along an integralcurve that "emerges" from point S and weassume that the initial segment of the curvecoincides with a segment of an exceptionalstraight line, the error may be minimizedif we allow for the convergence of integralcurves in the direction in which the valuesof sc diminish.Figure 72 illustrates the pattern of inte-gral curves in the vicinity of a singularpoint. The straight line passing throughpoint xt (the throat of the nozzle) corre-sponds to values at which the numeratorin the right-hand side of Eq. (156) vanishes,which points to the presence of extrema.2.10 Higher-Order Pointsof EquilibriumIn previous sections we studied the typesof singular points that emerge when theJacobian J * is nonzero. But suppose thatall the partial derivatives of the functions Xand Y in the right—hand sides of system (122)vanish up to the nth order inclusive. Thenin the vicinity of a singular point theremay be an iniinitude of phase—trajectorypatterns. However, if we exclude the pos-sibility of equilibrium points of the vortexand focal types emerging in this picture,then it appears that the neighborhood ofa singular point can be broken down into14*

212 Differential Equations in Applicationsa finite number of sectors belonging tothree standard types. These are the hyper-bolic, parabolic, and elliptic. Below we de-scribe these sectors in detail, but first let usmake several assumptions to simplify mat-ters.We assume that the origin is shifted tothe singular point, that is, :1:* = y* = O;the right-hand sides of system (122) canbe written in the formX (¤v» y) = X n (rv, y) + (D (re y)» (157)Y (rm y) = Ya (w, y) + ‘I’ (rn y>»where X ,, and Y,, are polynomials of de-gree n homogeneous in variables as and y(one of these polynomials may be identicallyzero), and the functions (D and W have inthe neighborhood of the origin continuousiirst partial derivatives. In addition, weassume that the functions<D(=·=» y) (Dx (¢» y) (Dy (=¤, y)(x2__‘_y2)(7l+1)/2 ’ (x2,_|_y2)7l·/2 ’ 1‘I’<=·=, y) ‘I’x(¤¤, y) Wy (rv, y)(x2_)_y2)('Tl+1)/2 ’ (x2_·_y2)71/P ’are bounded in the neighborhood of theorigin. Under these assumptions the follow-ing assertions hold true.(1) Every trajectory of the system ofequations (122) with right-hand sides of the(157) form that "enters” the origin alonga certain tangent touches one of the ex-

Ch. 2. Qualitative Methods 213ceptional straight lines specified by theequationwYn (an y) —— !/Xu (fe y) = 0- (158)Since the functions X ,, and Y,, are homo-geneous, we can rewrite Eq. (158) as anequation for the slope A = y/x. Then theexceptional straight line is said to be sin-gular ifXa (vs y) = Yum y) =0on this straight line. Some examples ofsuch straight lines are shown in Figure 62.The straight lines defined by Eq. (158) butnot singular are said to be regular.(2) The pattern of the phase trajectoriesof (122) in the vicinity of one of the tworays that "emerge" from the origin andtogether form an exceptional straight linecan be studied by considering a small disk(centered at the origin) from which weselect a sector limited by two radii lyingsufficiently close to the ray on both sidesof it. Such a sector is commonly known asa standard domain.More than that, in the case of a regularexceptional straight line, which corre-sponds to a linear factor in Eq. (158), thestandard domain considered in a disk ofa sufficiently small radius belongs t.o eitherone of two types: attractive or repulsive.(2a) The attractive standard domain ischaracterized by the fact that each tra-

214 Diflerential Equations in Applications

Fig. 74

Fig. 75jectory passing through it reaches theorigin along the tangent that coincideswith the exceptional straight line (Fig-ure 74).(2b) The repulsive standard domain ischaracterized by the fact that only onephase trajectory passing through it reachesthe singular point along the tangent thatcoincides with the exceptional straight line.All ot.her phase trajectories of (122) thatenter the standard domain through theboundary of the disk leave the disk bycrossing one of the radii that limit thedomain (Figure 75).

Ch. 2. Qualitative Methods 215Let us turn our attention to the followingfact. If the disk centered at the origin issmall enough, the two types of standarddomains can be classified according to thebehavior of the vector (X, Y) on theboundary of the domain. ’[`he behavior ofvector (X, Y) can be identified here withthe behavior of vector (X ny Yn). Morethan that, it can be demonstrated that if,as assumed, a fixed exceptional directiondoes not correspond to a multiple root ofthe characteristic equation, the vector con-sidered on one of the radii that limit thedomain is directed either inward or out-ward. Then if in the first case the vectorconsidered on the part of the boundaryof the standard domain that is the arc ofthe circle is also directed inward, and inthe second case outward, the standarddomain is attractive. But if the oppositesituation is true, the standard domain isrepulsive. It must be noted that in any casethe vector considered on the part of theboundary that is the arc of the circle isalways directed either inward or outwardsince it is almost parallel to the radius.Standard domains corresponding to sin-gular exceptional directions or multipleroots of the characteristic equation havea more complicated nature, but since tosome extent they constitute a highly rarephenomenon, we will not describe them here.lf we now turn, for example, to a saddle

216 Differential Equations in Applicationspoint, we note that it allows for fourrepulsive standard domains. In the neigh-borhood of a singular point of the nodaltype there are two attractive standarddomains and two repulsive.(3) If there exist real-valued exceptional-direction straight lines, the neighborhoodof a singular point can be divided into afinite number of sectors each of which isbounded by the two phase trajectories of(122) that "enter" the origin along definitetangents. Each of such sectors belongs toone of the following three types.(3a) The elliptic sector (Figure 76) con-tains an infinitude of phase trajectoriesin the form of loops passing through theorigin and touching on each side of theboundary of the sector.(3b) The parabolic sector (Figure 77)is filled with phase trajectories that connectthe singular point with the boundary of theneighborhood.(3c) The hyperbolic sector (Figure 78)is filled with phase trajectories that ap-proach the boundary of the neighborhoodin both directions.More precisely:(Zia) Elliptic sectors are formed betweentwo phase trajectories belonging to twosuccessive standard domains, both of whichare attractive.(4b) Parabolic sectors are formed betweentwo phase trajectories belonging to" two

Ch. 2. Qualitative Methods gnFig. 76 Fig. 77Fig. 78successive standard domains one of whichis attractive and the other repulsive. Allphase trajectories that pass through thelatter domain touch at the singular pointof the exceptional straight line that definesthe attractive domain.

218 Differential Equations in Applications(4c) Hyperbolic sectors are formed be-tween two phase trajectories belonging totwo successive repulsive standard domains.For example-, it is easy to distinguish thefour hyperbolic sectors at a saddle pointand the four parabolic sectors at a nodalpoint. Elliptic sectors do not appear inthe case of simple singular points, wherethe Jacobian J * is nonzero.lf a singular point does not allow for theexistence of real exceptional directions,the phase trajectories in its neighborhoodalways possess the vortex or focal struc-ture.*2.11 Inversion with Respectto a Circle and HomogeneousCoordinatesAbove we described methods for establish-ing the local behavior of phase trajectoriesof differential systems of the (122) type inthe neighborhood of singular points. Andalthough in many cases all required infor-mation can be extracted by following thesemethods, there may be a need to study the" Methods that make it possible to distinguishbetween a vortex {Joint and a focal point are dis-cussed, for examp e, in the book by V.V Ame1’-kin, N.A. Lukashevich, and A.P. Sadovskii,Nonlinear Vibrations in Second—0rder Systems(Minsk: Belorussian Univ. Press, 1982) (in Rus-sian .

Ch. 2. Qualitative Methods 219trajectory behavior in infinitely distantparts of the phase plane, as x2 —1~ y2 —> oo.A simple way of studying the asymptoticbehavior of the phase trajectories of thedifferential system (122) is to introducethe point at infinity by transforming theinitial differential system in an appropriatemanner, say by inversion, which is definedby the following formulas:.__ E. __ n{L- {32+*12 ’ y- §”+n“$ y(g2··`_;§··_F§i·,Geometrically this transformation consti-tutes what has become known as inversionwith respect to ci circle and maps the origininto the point at infinity and vice versa.Transformation (159) maps every finitepoint M (az, y) of the phase plane into pointM ’ (§, 1]) of the same plane, with points Mand M' lying on a single ray that emergesfrom the origin and obeying the conditionOM >< OM' =-· rz (Figure 79). It is wellknown that such a transformation mapscircles into circles (straight lines are con-sidered circles passing through the pointat infinity). For one thing, straight linespassing through the origin are invariantunder transformation (159). Hence, theslopes of asymptotic directions are theslopes of tangents at the new origin § =1] = O. Note that in the majority of

220 Differential Equations in Applicationsml ·¤wv, fFig. 79NFig. 80cases the new origin serves as a singularpoint. The reasons for this are discussedbelow.

Ch. 2. Qualitative Methods 221As for the question of how to constructin the old (sc, y)-plane a curve that has adefinite asymptotic direction, that is, hasa definite tangent at the origin of the new(Q, 1])-plane, this can be started by con-sidering the (§, ij)-plane, more precisely,by considering this curve in, say, a unitcircle in the (Q, 1])-plane. The fact is thatsince a unit circle is mapped, via transfor-mation (159), into itself, we can always estab-lish the point where the curve intersects therespective unit circle in the (1:, y)-plane;any further investigations can be carriedout in the usual manner.We also note that completion of the(:1:, y)-plane with the point at infinity istopologically equivalent to inversion ofthe stereographic projection (Figure 80),in which the points on a sphere are mappedonto a plane that is tangent to the sphereat point S. The projection center N is theantipodal point of S. It is clear that theprojection center N corresponds to thepoint at infinity in the (41:, y)-plane. Con-versely, if we map the plane onto thesphere, a vector field on the plane trans-forms into a vector field on the sphere andthe point at infinity may prove to be a sin-gular point on the sphere.Although inversion with respect to acircle is useful, it proves cumbersome andinconvenient when the point at infinity hasa complicated structure. In such cases

222 Differential Equations in Applicationsanother, more convenient, transformation ofthe (:1:, y)—plane is used by introducinghomogeneous coordinates:as = §/z, y = 1]/z.Under this transformation, each point ofthe (x, y)-plane is associated with a tripleof real numbers (§, 1], z) that are notsimultaneously zero and no difference ismade between the triples (§, eq, z) and(k§, lm, kz) for every real k q'-= O. If apoint (x, y) is not at infinity, z =;·‘= 0. Butif z = 0, we have a straight line at infinity.The (x, y)-plane completed with the straightline at infinity is called the projective plane.Such a straight line may carry several sin-gular points, and the nature of these pointsis usually simpler than that of a singularpoint introduced by inversion with respectto a circle.If we now consider a pencil of lines anddescribe its center with a sphere of, say,unit radius, then each line of the pencilintersects the sphere at antipodal points.This implies that every point of the pro-jective plane is mapped continuously andin a one-to-one manner onto a pair of anti-podal points on the unit sphere. Thus, theprojective plane may be interpreted as theset of all pairs of antipodal points on a unitsphere. To visualize the projective plane,we need only consider one-half of the sphereand assume its points to be the points of the

Ch. 2. Qualitative Methods 233é‘·=z=·—0**0 AM;we ¢=z=0lL i iA'E w.9Fig. 81projective plane. If we orthcgonally projectthis hemisphere (say, the lower one) ontothe ot-plane, which touches it at pole S(Figure 81), the projective plane is mappedonto a unit disk whose antipodal pointson the boundary are assumed identical.Each pair of the antipodal points of theboundary corresponds to a line at infinity,and the completion of the Euclidean planewith this line transforms the plane intoa closed surface, the projective plane.2.12 Flow of ra Perfect GasThrough a Rotating Tubeof Uniform Cross SectionIn some types of turboprop helicopters andairplanes and in jet turbines, the gaseousfuel—air mixture is forced through rotating

224 Differential Equations in ApplicationsdCK 5 C ’ VQH `eFig. 82tubes of uniform cross section installedin the compressor blades and linked by ahollow vertical axle. To establish the opti-mal conditions for rotation we must analyzethe flow of the mixture through a rotatingtube and link the solution with boundaryconditions determined by the tube design.In a blade the gaseous mixture participatesin a rotational motion with respect to theaxis with constant angular velocity coand moves relative to the tube with anacceleration v (dv/dr), where v is the speedof a gas particle with respect to the tube,and r is the coordinate measured along therotating compressor blades.Figure 82 depicts schematically a singlerotating tube of a compressor blade.* It is* See I . Kestin and S.K. Zaremba, "Adiabaticone-dimensional flow of a perfect gas through arotating tube of uniform cross section," Aeronaut.Quart. 4: 373-399 (1954).

Ch. 2. Qualitative Methods 225assumed that the fuel-air mixture, whoseinitial state is known, is supplied alongthe hollow axle to a cavity on the axle inwhich the flow velocity may be assumedinsignificant. The boundary of the cavityoccupied by the gas is denoted by a, thegas is assumed perfect with a speciiic heatratio y, and all the processes that the gasmixture undergoes are assumed reversiblyadiabatic (exceptions are noted below).It is assumed that the gas expands througha nozzle with an outlet cross section b thatat the same time is the inlet of a tube ofuniform cross section. The expansion ofthe gas mixture from state a. to state bis assumed to proceed isentropically; wedenote the velocity of the gas after expan-sion by vi and the distance from the rotationaxis O, to the cross section b by rl.When passing through the tube, whoseuniform cross—sectional area will be denotedby A and whose hydraulic diameter by D,the gas is accelerated thanks to the com-bined action of the pressure drop and dy-namical acceleration in the rotation compres-sor blade. We ignore here the effect producedby pressure variations in the tube (if theyexist at all) and by variations in pressuredrop acting on the cross section plane, bothof which are the result of the Coriolis force.The last assumption, generally speaking,requires experimental verification, sincethe existence of a lateral pressure drop mayis-0770

226 Differential Equations in Applicationsserve as a cause of secondary flows. But ifthe diameter of the tube is small comparedto the tube’s length, such an assumptionis justified.It is now clear that the equations of mo-mentum and energy balance for a compres-sible mixture traveling along a tube ofuniform cross section must be modified soas to allow for forces of inertia that appearin a rotating reference frame. As for thecontinuity equation, it remains the same.Now let us suppose that starting fromcross section c at the right end of the tubethe gas is compressed isentropically andpasses through a diverging nozzle. In theprocess it transfers into a state of rest withrespect to the compressor blade in the secondcavity at a distance r3 from the rotationaxis and reaches a state with pressure Pdand temperature Td.From the second cavity the gaseous mix-ture expands isentropically into a converg-ing or converging—diverging nozzle in sucha manner that it leaves the cavity at rightangles to the tube’s axis. This producesa thrust force caused by the presence ofa torque.Below for the sake of simplicity we as-sume that the exit nozzle is a convergingone and has an exit (throat) of area A*.Denoting the external (atmospheric) pres-sure by Pa, we consider two modes of pas-sage of the mixture through the nozzle.

Ch. 2. Qualitative Methods 227The first applies. to a situation in whichthe P3-to-Pd ratio exceeds the criticalvalue, orP.,/Pd > (2/(v + i))""""’·In this case the flow at the nozzl0°s throatis subsonic and, hence, pressure P3 at thethroat is equal to the atmospheric pressure,that is, P3 —-= P3. The second case appliesto a situation in which the Pa-to-Pd ratiois below the critical value, orP.,/Pa < (2/(v + i)"’"‘*’-The pressure P3 at the nozzle’s throat hasa fixed value that depends on Pd but noton P3. Hence,P3 -= (2/(V + i))""""’ Pa-In the latter case the flow at the nozzle’sthroat has the speed of soundUs = (2/(v + i))"2 aawhere ad depends only on temperature Td.In further analysis the flow is assumedadiabatic everywhere and isentropic every-where except in the tube between cross sec-tions b and c.As in the case where we derive the differ-ential equation that describes the adia-batic flow of a perfect gas through a nozzleof varying cross section, let us now considerthe continuity equation, the equation of15*

228 Differential Equations in ApplicationsxU 7 U+d0,0 P+6Z'/Dv /[.,4r' + drFig. 83momentum, and the equation of energybalance. For instance, the equation of con-tinuity in this case has the form- .2. ..1; .4; _1p— V .. V3 A ..const, (160)with V the speciiic volume and xp the fluxdensity, or‘\|> = mr/A,where m' is the flux mass.To derive the equation of motion, or themomentum equation, we turn to Figure 83.We note that the dynamical effect of therotational motion of the compressor bladecan be used to describe the flow with re-spect to the moving tube, where in accord-ance with the D’A1embe1·t principle theforce of inertiadI = {3- wzf dr

Ch. 2. Qualitative Methods 229is assumed to act in the positive directionof r. Hence, the element of mass dm 2(A/V) dr moves with an accelerationv(dv/dr) caused by the combined actionof the force of inertia dI, the force of pres-sure A dP, and the friction force dF =(4A/D) dr. Here ·t : Z. (02/2V), where 7.depends on the Reynolds number R. Asa first approximation we can assume that2. remains constant along the entire tube.With this in mind, we can write the mo-mentum equation in the formA dv 2}»Av2 A·I7dT°U·H·;·-—-· dT°·l······I7ZU2T°dr,or after certain manipulations,V dP —|—v dv-}- ED?-L v2dr—w2r dr= 0. (161)As for the energy-balance equation, it issimple to derive if we use the first law ofthermodynamics for open systems and bearin mind that the amount of work performedby the system is wzr dr. Thusdh—|—vdv—-w2rdr=O,where h is the enthalpy. If we define thespeed of sound a via the equationh:`QlT>we find thatada-l——·Y—%vdv—— -3%-}—w2rdr=O.

230 Differential Equations in ApplicationsThis leads us to the following energy-balanceequation:ag Z az -1- -Y-gi U2- 3% ww. (162)If we now introduce the dimensionlessquantitiesM0 = v/ao, as = r/D, G2 = w2D”/a§, (163)we arrive at the equation2 -1 -1%:1—-JT-M§-l- %—G2x2. (164)By excluding the pressure and specificvolume from the basic equations (160),(161), and (164) we can derive an equationthat links the dimensionless quantity M0with the dimensionless distance. This equa—tion serves as the basic equation for solvingour problem. The continuity equation (160)implies thatV = v/wp. (165)Then, combining__ ___ 'yPvG2 -—- — Twith Eq. (162), we arrive at the followingrelationships:_ a2 _ ·y—1 _ ’y—1 wzpr (Ri W“‘*”wT"2)‘f»dP _ Y—1 wgr __ y——1 a2YT"` y v ( 2v Jr qe;2v—1 w2r2 dv ..+ ig" *,7) 7;; · (166)

Ch. 2. Qualitative Methods 231Substituting the value of V from Eq. (165)and the value of dP/dr from Eq. (166) intothe momentum equation (161) and allowingfor the relationship (163), we arrive at thedifferential equationdjtlg Z 2M1§(22,yM§—-Gaz;) o (167)at 4-% Mg .!. lg.; GzxzAssuming that1M§=y, m=2?»v, p=—y2—(v+1>.1q : if (Y '- Uswe can rewrite Eq. (167) asdy __ 2y (my——G2x)"ds?" 1——py—|—qG2x2 ° (*68)The differential equation (168) is theobject of our further investigation. In-troducing the variables (159), we reduce it toQ Z 2E*l(E·°··P*lE+¢1G“E,2)+2*1(*12·~E2)Qdi A (E“—·PT\E ·|· <1G”§’) + @.*129 ’(169)where E : Q2-}- q2, Q = mq- G2§, A : §,2·····T]2.Clearly, the origin constitutes a singularpoint for Eq. (169). The lowest—order termsin both numerator and denominator arequadratic. If we discard higher-order terms,

232 Differential Equations in Applicationsas we did on p. 211 when discussing higher-order equilibrium points, we iind thatY4 (gv Tl)= 2qG2E211 + 211 (112 — E2) (mn — G2E),X4 (gv= qG2E2 (E2 — 112) + 4E112 (mn — G2E),and, hence, the origin is a higher-ordersingular point.The characteristic equation of the differ-ential equation (169) assumes the followingform after certain manipulations:En (E2 + 112) [(q + 2) G2E —· 2*1111] = 0-(170)This leads us to three real—valued excep-tional straight lines:(¤) E=0» (b)11 =0»<¤> (q + 21 Gag — 2m. = 0. (27*)Each is a regular straight line and corre-sponds to one of the factors in Eq. (170).It is easy to see that for § and 1] positivethe value of X4 (§, iq) is positive in theneighborhood of all three exceptionalstraight lines, with the result that theexpressionYi (E, 11) _;X4 (gs gZ gl} TIS) l(q"l`2) G2E"2”"llE lqG2E2 (E2 — 112)+ 4En* (mn —- G2E)l

Ch. 2. Qualitative Methods 233has the same sign in both the first quadrantand the neighborhood of the straight linesas the left-hand side of (171c) and, hence,is negative below the straight line (171c)and positive above. The explanation liesin the fact that (172) may change sign onlywhen an exceptional straight line is crossed.Geometrically this means that the right-hand side of the differential equationEl; Y4 (gv ll)dg X4 (gsdetermining the behavior of integral curvesin a disk of sufficiently small radius andcentered at the origin fixes an angle greaterthan the angle of inclination of the radiithat lie in the first quadrant between thestraight lines (171b) and (171c). As forthe radii lying between the straight lines(171c) and (171a), on them the right—handside of this differential equation fixes anangle that is smaller than the angle ofinclination of these radii (Figure 84). Thus,the standard domain containing the straightline (171c) must be repulsive, while thedomains containing the straight lines (17ia)and (171b) must be attractive. In view ofthe symmetry of the field specified by vector(X4, Y4), the above facts remain valid forstandard domains obtained from those men-tioned earlier by a rotation about thesingular point through an angle of 180°.

234 Differential Equations in Applications/\em 7 X—¤~ f`\ TY ’ //4// K// x \jr +\Ai `é\/7 EvFig. 84Thus, there are exactly two integralcurves that "enter" the singular point alongthe tangent (171c) and an infinitude ofintegral curves that touch the coordinateaxes (171a) and (171b) at the point of rest.We see then that the second and fourthquadrants contain elliptic sectors sincethey lie between two successive attractivestandard domains (Figure 85). Each of thefirst and third quadrants is divided intotwo sectors by the integral curves thattouch the exceptional straight line (171c)at the origin. These sectors are parabolicbecause they lie between two successivestandard domains, one of which is attractiveand the other repulsive. More than that,all the integral curves except those thattouch the straight line (171c) touch the

Ch. 2. Qualitative Methods 2357isFig. 85coordinate axes (171a) and (17lb) at thesingular point.Basing our reasoning on physical con-siderations, we can analyze the differentialequation (168) solely in the first quadrantof the (sc, y)—plane. Turning to this plane,we see that there exists exactly one integralcurve having an asymptotic direction witha slope (q —{- 2) G2/2m. All other integralcurves allow for the asymptotic directionof one of the coordinate axes. Indeed, it iseasy to prove that all these integral curvesasymptotically approach one of the coor-dinate axes, that is, along each of them inthe movement toward infinity not only doesy/x —>0 or :1:/y -—>0 but so does y -+0 orcc —->0 (with .2: -—>oo or y ->cx>, respec-

236 Diflerential Equations in Applications

0.2:Fig. 86tively). This is illustrated in Figure 86,where some integral curves have been plot-ted at great values of :1: and y. Note that thepattern of integral curves at a finite dis-tance from the origin depends primarilyon the position of the singular points, and·a special investigation is required to clarifyit. It can also be proved that the integralcurves that approach the coordinate axesasymptotically (Figure 86) leave the firstquadrant at a finite distance from theorigin.2.13 Isolated Closed TrajectoriesWe already know that in the case of asingular point of the vortex type a certainregion of the phase plane is completelyfilled with closed trajectories. However,

Ch. 2. Qualitative Methods 237a more complicated situation may occurwhen there is an isolated closed trajectory,that is, a trajectory in a certain neigh-borhood of which there are no other closedtrajectories. This case is directly linkedwith the existence of isolated periodicsolutions. Interestingly, only nonlinear dif-ferential equations and systems can haveisolated closed trajectories.Isolated periodic solutions correspondto a broad spectrum of phenomena and pro-cesses occurring in biology, radiophysics,oscillation theory, astronomy, medicine,and the theory of device design. Such solu-tions emerge in differential models in eco-nomics, in various aspects of automaticcontrol, in airplane design, and in otherfields. Below we study the possibility ofisolated periodic solutions emerging in pro-cesses that occur in electric circuits; we alsoconsider as a model the nonlinear differ-ential system{j—§°—=—y+=v(i ——·w”—y2).dy (173)Ei-:-:z:—|—y(l —-0:2-y2).To solve this system, we introduce polarcoordinates r and 9, where x = rcos 6and y = rsin 9. Then, differentiating therelationships x2 —|— y2 == rz and 9 =

238 Differential Equations in Applicationstan·1(y/x) with respect to t, we arrive atdx dy __ dr dy _ dx _ _ dB`"w+ya7··rw· xr; ff aT··’2‘a2‘·(174)Multiplying the first equation into sc andthe second into y, adding the products,and allowing for the first relationship in(174), we find thatdrr -8-2-: rz (1 -1*2). (175)Multiplying the second equation in (173)into sc and the first into y, subtracting oneproduct from another, and allowing for thesecond relationship in (174), we find thatrz gil; rz. (176)System (173) has only one singular point,O (0, 0). Since at the moment we are onlyinterested in constructing trajectories, wecan assume that r is positive. Then Eqs.(175) and (176) imply that system (173)can be reduced to the formdr df) _717--T(1··—T°2), ·aT—1.Each of these equations can easily be inte-grated and the entire family of solutions,

Ch. 2. Qualitative Methods 239as can easily be seen, is given by the for-mulas1’ 9:-1t+t°’ (178)or, in terms of the old variables x and y,COS + tg) sin to)x = -·T ———— » I! = +*——_————-——‘V 1+C¢‘”‘ I/1+ 06**If now in the first equation in (178) we putC-:0, we get r=1 and 6:-—t-{—t0.These two relationships define a closedtrajectory, a circle x2 —l- y2 L-= 1. If C isnegative, it is clear that r is greater thanunity and tends to unity as t —> +oo. Butif C is positive, it is clear that r is less thanunity and tends to unity as t -» +oo. Thismeans that there exists only one closedtrajectory r == _1 which all other trajectoriesapproach along spirals with the passage oftime (Figure 87).Closed phase trajectories possessing suchproperties are known as limit cycles or,more precisely, (orbitally) stable limit cycles.In fact, there can be two additional typesof limit cycles. A limit cycle is said to be(orbitally) unstable if all neighboring tra-jectories spiral away from it as t—»—|—o<>.And a limit cycle is said to be (orbitally)half-stable if all neighboring trajectorieson one side (say, from the inner side) spiralon to it and all neighboring trajectories

240 Differential Equations in Applications.77.z*Fig. 87on the other side (say, on the outer side)spiral away from it as t->-|—¤¤.In the above example we were able tofind in explicit form the equation of a closedphase trajectory, but generally, of course,this cannot be done. Hence the importancein the theory of ordinary differential equa-tions of criteria that enable at least spe-cifying the regions where a limit cyclemay occur. Note that a closed trajectory of(122), if such a trajectory exists, containswithin its interior at least one singularpoint of the system. This, for one thing,implies that if there are no singular pointsof a differential system within a region ofthe phase plane, there are no closed tra-jectories in the region either.

Ch. 2. Qualitative Methods 241T1f/.·••"""°`_—"-°`7<\\X 0 ` F’ x( iI 1l 1\ /\\ //Fig. 88Let D be a bounded domain that liestogether with its boundary in the phaseplane and does not contain any singularpoints of system (122). Then the Poincaré-Bendixson criterion holds true, namely, ijI` is a trajectory of (122) that at the initialmoment t:-· to emerges from a point thatlies in D and remains in D for all t) to,then F is either closed or approaches a closedtrajectory along a spiral with the passage oftime.We illustrate this in Figure 88. Here Dconsists of two closed curves I`, and I`2and the circular domain between them.With each boundary point of D we associatea vectorV(¢¤» y) = X (rv, y)i+Y(¤¢. y)i-Then, if a trajectory T` that emerges at theinitial moment t= to from a boundarypoint, enters D, and remains there at all16-0770

242 Differential Equations in Applicationsmoments tgto, then, according to theabove criterion, it will along a spiralapproach a closed trajectory F0 that liesentirely i11 D. The curve I`0 must surrounda singular point of the differential system,point P, not lying in D.The differential system (173) providesa simple example illustrating the appli-cation of the above criterion in finding limitcycles. Indeed, system (173) has only onesingular point, O (O, O), and, therefore,the domain D lying between the circleswith radii r = 1/2 and r = 2 contains nosingular points. The first equation in (177)implies that dr/dt is positive on the innercircle and negative on the oute1·. Vector V,which is associated with the points on theboundary of D, is always directed into D.This means that the circular domain lyingbetween circles with 1·adii r = 1/2 andr : 2 must contain a closed trajectory of thedifferential system (173). Such a closedtrajectory does indeed exist, it is a circle ofradius r = 1.Note, however, that great difficulties aregenerally encountered in a system of the(122) type when we wish to realize prac-tically the Poincaré-Bendixson criterion,since no general methods exist for buildingthe appropriate domains and, therefore,success depends both on the type of systemand on the experience of the researcher. Atthe same time we must bear in mi11d that

Ch. 2. Qualitative Methods 243finding the conditions in which no limitcycles exist is no less important than estab-lishing criteria for their existence. ln thisrespect the most widespread condition is theDulac criterion: if there exists a functionB (x, y) that is continuous together with itsfirst partial derivatives and is such that ina simply connected domain D of the phaseplane the sum0 (BX) 0 (BY)0.2: + Oyis a function of fixed sign,* then no limitcycles of the dijjerential system (122) canexist in D. At B (x, y) Ei 1 the criteriontransforms into the Bendixson criterion.lf we turn to the differential equation(156), which constitutes a differential modelfor describing an adiabatic OIl9"(llI1"l|3l1Sl(l113lflow of a perfect gas of constant specificheat ratio througli a nozzle with drag, thenfor this equation we haveX(¤v. y)=(i——y)F(¤¤),...1 IY(¤v, y)=4y (1l+ y) <vqy—F (·¤¤))·If we putv—1 ·*Btw, y)= {4yF(¤v) [1+ —q,—— 0]}* That is, positive definite or negative definite.16*

244 Difierential Equations in Applicationswe find that6(BX) 6(BY) _ Yqdx + dy = F(x) >Oand, hence, Eq. (156) has no closed integralcurves.Let us discuss one more concept that canbe employed to establish the existence oflimit cycles. The concept is that of theindex of a singular point.Let I` be a simple closed curve (i.e. acurve without self~intersections) that isnot necessarily a phase trajectory of system(122), lies in the phase plane, and does notpass through the singular points of thissystem. Then, if P (x, y) is a point of I`,the vecto1·V(¤v,y)=X(¤v» 1/)i+Y(¤>,y)i.with i and j the unit vectors directed alongthe Cartesian axes, is a nonzero vector and,hence, is characterized by a certain directionspeciiied by an angle 9 (Figure 89). Ifpoint P (x, y) moves along I`, say, counter-clockwise a11d completes a full cycle, vectorV performs an integral number of cyclesin the process, that is, angle 9 acquires anincrement A9 = 2nn, with n a positiveinteger, a negative integer, or zero. Thisnumber n is said to be the index of the closedcurve I` (or the index of cycle I`).lf we begin to contract I` in such a mannerthat under this deformation I` does not

Ch. 2. Qualitative Methods 245Vy 0F0 eZ`Fig. 89pass through singular points of the givenvector Held, the index of the cycle must,on the one hand, vary continuously and,on the other, remain an integer. This meansthat under continuous deformation of thecurve the index of the cycle does not change.This property leads to the notion of theindex of a singular point as the index of asimple closed curve surrounding the sin-gular point.The index has the following properties:(1) the index of a closed trajectory of thediyjerential system (122) is equal to +1,(2) the index of a closed curve surroundingseveral singular points is equal to the sumof the indices of these points, and(3) the index of a closed curve encompassingonly ordinary points is zero.

246 Differential Equat·ions in ApplicationsThis implies, for one thing, that sincethe index of a closed trajectory of system(122) is always +1, a closed trajectory mustenclose either one singular point with anindex -}-1 or several singular points witha net index equal to -{-1. This fact is oftenused to prove the absence of limit cycles.The index of a singular point is calculatedby the formulan:1+§#, uw)where e is the number of elliptic sectors,and h is the number of hyperbolic sectors.For practical purposes the following simplemethod may be suggested. Suppose that Lis a cycle that does not pass through sin-gular points of (122) and is such that anytrajectory of (122) has no more than a finitenumber of points common to L. The tra-jectories may intersect L or touch it. Inthe latter case only exterior points of con-tact (type A) or interior points of contact(type B) are taken into account, while theC-type points, points of inflection, are not(see Figure 90). We can still use formula(179) to calculate the index of a singularpoint, but now e is the number of interiorpoints of contact and h the number ofexterior points of contact of trajectoriesof (122) with cycle L.Figure 91 depicts singular points with

Ch. 2. Qualitative Methods 247OamQ MB'<\>LFig. 90indices 0, +2, +3, +1, and -2, re-spectively.Earlier it was noted that constructinga complete picture of the behavior of thephase trajectories of the differential system(122) is facilitated by introducing the pointat infinity via transformations (159). To-pological considerations provide a verygeneral theorem which states that whena continuous vector field with a finite num-ber of singular points is specified on asphere, the not index of the points is +2.Thus, if the net index of all the singularpoints of a differential system (possessinga finite number of such po.ints) that lie ina finite phase-plane domain is distinct_from +2, the point at infinity must be asingular point with a nonzero index.But if instead of inversion we employhomogeneous coordinates, the net index ofall the singular points is already +1. That

Ch. 2. Qualitative Methods 249this is so can be seen from the fact that ifthe plane is projected onto a sphere withthe center of projection placed at the centerof the sphere, two points on the sphere cor-respond to a single point on the projectiveplane, and the circumference of the greatcircle parallel to the plane corresponds toa straight line at infinity.If we turn to Eq. (168), which describesadiabatic one-dimensional flow of a perfectgas through a rotating tube of uniform crosssection, then, as shown in Figure 85, for thesingular point at infinity we have e = 2and h == O. It follows from this that theindex of this point is +2 and does notdepend on the values of the constants inEq. (168). This implies, for one thing, thatthe net index of finite singular points iszero. It can also be shown that, dependingon whether the straight line specified bythe equation my - Ganz == 0 has two points,one double point, or not a single point ofintersection with the parabola fixed by theequation 1 -—py + qG2x2 = O (which, in turn,is equivalent to G being greater than G0,equal to G0, or less than G0, with G0 =2mq*/2/p), the following combinations offinite singular points occur.(a) G > G0. A saddle point and a nodalpoint.(b) G = G0. A higher-order singular pointwith two hyperbolic (h —.= 2) sectors andtwo parabolic (e = O).

250 Differential Equations in Applications(c) G <G0. Singular points are absent.We see that in three cases the net indexis zero, as it should be.2.14 Periodic Modes in ElectricCircuitsWe will show how limit cycles emerge ina dynatron oscillator (Figure 92). An anal-ysis of the operation of such an oscillatorleads to what is known as the Van der Polequation. Although phenomena linked withgeneration of limit cycles can be illustratedby examples from mechanics, biology, andeconomics} we will show how such phe-nomena emerge in the study of electriccircuits.Figure 92 depicts schematically a dyna-tron oscillator, with the ia-va characteristicshown by a solid curve in Figure 93. Hereia is the current and va the voltage in theté ' r Ce '|' +r...L__Fig. 92

Ch. 2. Qualitative Methods 251screen-grid tube, the dynatron. The circuitconsists of resistance r, inductance L,and capacitance C connected in paralleland known as the tank circuit, which isconnected in series with the screen—grid tube.The real circuit can be replaced in this casewith an equivalent circuit shown schematic-ally in Figure 94. The characteristic of thetube may be approximated with a third-degree polynomial i = ow + YU3, which isshown by a dashed curve in Figure 93.Here i and v stand for the coordinates ina system whose origin is shifted to thepoint of inflection O. As follows fromFigure 93,on > O, Y > O.In accordance with one of Kirchhoff’s laws,i·l·ir+iL·l—io=O,with ir = v/r, L (<liL/dt) == v, and ic =Cv. As a result of simple manipulations wearrive at the following differential equation:" <¤ 1 3v 2 1-;..." +l·zr+‘;z‘+‘@"" lvi LC -*0-If we now putcz,. 1 _ 3y __ _1___ 2‘E·‘+‘?F"“¤ 2·`“b¤ rc —"”<··we can write the p1·evious equation in theformall- _(;z —l- bvz) + o>§v = O.

252 Differential Equations in Applicationsia li /* /{ //I-\\\ 1 /- # .... -:'--- -7/ -,./ 0| / U/ l \\ 1/’ n’ 1IOd vaFig. 93i-•(-—-—-——l #1 it rlif R f" A CFig. 94This differential equation is known as theVan der Pol equation. If we introduce thetransformations• ••"=y· ""yW=we can associate with the Van der Pol

Ch. 2. Qualitative Methods 253equation the following first-order differen-tial equation:dy ___ __ (¢+bv2)y+<¤3v __ Y(v. y)TY" y "` V (v. y) (180)The only finite singular point of thisequation is the origin, andJ* (0, 0) = m§>0,D (v, y) == —-(a -4- bvz).Since b>0, we assume that a>0 andconclude that divergence D does not re-verse its sign and, hence, Eq. (180) canhave no closed integral curves. We, there-fore, will consider only the case where ais negative, that is, ot< —-1/r. This im-plies that D (0, 0) == —a > 0 and, hence,the singular point is either a nodal pointor a focal point. lf we now consider thedifferential system corresponding to thedifferential equation (180), that is,dy __ dv __`EZ""Y(Uv y)v `(`E"·‘I/(vv y)vwe see that as t grows, the representativepoint moves along a trajectory toward thesingular point. Thus, a trajectory thatemerges from the point at infinity cannotreach the singular point at the origin nomatter what the value of t including thecase where t-= —l-oo. This implies that ifwe can prove that a trajectory originatingat the point at infinity resembles a spiral

254 Differential Equations in Applicationswinding onto the origin as t-><x>, thiswill guarantee that at least one limit cycleexists.The existence proof for such a cycle canbe obtained either numerically or analyti-cally. A numerical method suggested by theDutch physicist and mathematician Vander Pol (1889-1959) consists in buildinga trajectory that originates at a pointpositioned at a great distance from theorigin and in checking whether this tra-jectory possesses the above-mentioned prop-erty. Such a procedure yields an app1·oxi-mation to the limit cycle, but it can beemployed only in the case where concretenumerical values are known.Below we give a proof procedure* basedon analytical considerations and the in-vestigation of the properties of singularpoints at infinity. Here, in contrast to themethod by which Eq. (168) was studied,we employ the more convenient (in thepresent problem) homogeneous-coordinatetransformationsv = E,/z, y = 1]/z. (181)The straight line at infinity is fixed by theequation z = O. To reduce the number of* See the paper by J. Kestin and S.K. Zaremba,"Geometrical methods in the analysis of ordinarydifferential equations," Appl. Sei. Res. B3: 144189 (1953).

Ch. 2. Qualitative Methods 255variables to two, we first exclude the pointQ = z = 0. This can be done if we assumethat § = 1. Thenv == 1/z, y : 1]/z.In this case the differential system asso-ciated with Eq. (180) is transformed todz __ dn (az2—|—b)·q—l-0J?z2717*- " "l·" · af: **2-It is convenient to introduce a new para-meter, 9, in the following manner:dt == zz dt). (182)Then this system of equations (3311 be writ-ten asgg: —-(azz-}—b)1]—c0§z`·*—1]2z2=i’!_,d (183)HT)-- T|Z —— .Note, {irst, that the straight line at infinity,.z -= 0, constitutes a trajectory of the dif-ferential system (183) and as t grows (hence,as 9 grows) the representative point movesalong this trajectory toward the only sin-gular point z = iq = O. The characteristicequation, which in this case has the form—b1jz = O, speciijes t]1e regular exceptionaldirection z = O, with two repulsive regionscorresponding to this direction, which canbe establishedby studying the appropriate

256 Diflerential Equations in ApplicationszQT I E3:0\112/ x\\’?/\ /j/4)//"Z‘ Jr EFig. 95vector field (Figure 95). The second excep-tional direction 1] = 0 is singular and,therefore, additional reasoning is required.The locus of points 9 == U is a curve thattouches the straight line iq == 0 at theorigin and passes through the second andthird quadrants (Figure 95), which makesit possible to fix three directions, I, II,and III, on different sides of the symmetryaxis z = O. The region lying between thecurve fixed by the equation 9 = 0 andthe axis fr] = O is topologically equivalentto two repulsive regions. Thus, on each sideof the straight line mq = 0 there is at least

Ch. 2. Qualitative Methods 257one t1·ajectory that touches the straightline at the origin.If we now consider the differential equa-tiondn __ a b wt ii ._jy;—7+;;+7,;+·;—f(¤» Z),we note that for small values of | TII0j __ 1 m?,ia-? (ir?) <"in the second quadrant between curve ay =O and axis fr] = O. Hence, if we take twophase trajectories with the same value of 1],it is easy to see that the representativepoints moving along these trajectories willalso move apart as z decreases. This meansthat on each side of the straight line 1] = Othere can be only one phase trajectory thattouches this straight line at the originand belongs to the region considered. lt isalso clear that the qualitative picture issymmetric about the axis z == O. More thanthat, since for small values of z and positivetj we havela/2 I2 wi/luz I.t.here can be no curves that touch the axis1] = 0 at the origin and pass through thefirst or fourth quadrant. Such curves arealso absent from the region that lies to theleft of the curve fixed by the equationiy = O since in this case 9 is positive andl7—U770

258 Differential Equations in ApplicationsTZ has the same sign as z (Figure 95,arrows IV). This reasoning suggests thatthe singular point rj = z == O is a saddlepoint. The two phase trajectories that reachthis point as t —> 5}-oo are segments of thestraight line at infinity, z = O, that link theprevious point with the point 7; = z = 0.The other two phase trajectories reach thesaddle point as t-—>-—· oo.In ou1· reasoning we did not discuss thepoint lf; = z = O. To conclude our investi-gation, let us put 1] = 1 in (181). Then thedifferential system associated with the dif-ferential equation (180) can be written asfollows:.9§_..Zz_{_ré(aZz·bz 2 2..;do- + §+w,,§Z)—-1,?;·=Z (¤Z2+b§2+w§§Z2)=Q,where the variable 9 is defined in (182). Thepoint E, == z = U proves to be singula1·, andthe characteristic equation here is z3 = 0.Hence, the exceptional direction z = O issingular, too. The curve fixed by the equa-tion P (E, z) = Ol (Figure 96) touches theaxis z = O at the origin and has a cuspidalpoint there, while the curve fixed by theequation Q (§, z) = 0 has a multiple pointat the origin. By studying the signs of Pand Q we can establish the pattern of thevector iield (Figure 96) and the phase-tra-

rCh. 2. Qualitative Methods 2590:0 Z //E /x\i\ \ \ / /"/ /’/ \ §xy // \\ \..4.. iv-/ \\\\Fig. 96\ z//\\ // g’( ,` t/ - \\\\\Fig. 97jectory pattern in the neighborhood of thesingular point ig = z = O (Figure 97).We note, for one thing, that far from theaxis ig': O there are no phase trajectoriesthat enter the singular point from the right.17*

260 Differential Equations in ApplicationsThis follows from the fact that in the firstand fourth quadrantsIZ/E I> IQ/P I-The above reasoning shows that the Vander Pol equation has no phase trajectoriesthat tend to infinity as t grows but has aninfinitude of phase curves that leave infinityas t grows. This proves the existence of atleast one limit cycle for the Van der Poldifferential equation.2.15 Curves Without ContactIn fairly simple cases the complete patternof the integral curves of a given differentialequation or, which is the same, the phase-trajectory pattern of the correspondingdifferential system is determined by thetype of singular points and closed integralcurves (phase trajectories), if the latterexist. Sometimes the qualitative picturecan be constructed if, in addition to estab-lishing the types of singular points, we canfind the curves, separatrices, that link thesingular points. Unfortunately, no generalmethods exist for doing this. Hence, it isexpedient in qualitative integration toemploy the so—called curves without con-tact. The reader will recall that a curve oran arc of a curve with a continuous tangentiS said to be a curve (arc) without contactif it touches the vector (X, Y) specified

Ch. 2. Qualitative Methods 261by the differential system (122) nowhere.The definition implies that vector (X, Y)must point in one direction on the entirecurve. Thus, a curve without contact maybe intersected by the phase trajectories of(122) only in one direction when t growsand in the opposite direction when t di-minishes. Therefore, knowing the respectivecurve may provide information about thepattern of a particular part of a phase tra-jectory.Various auxiliary inequalities can beused in qualitative integration of differen-tial equations. Fo1· example, if two differ-ential equations are known, say,g'g";f(·Tv y)v 'g%`T—“g(xv y)and it is known that f(;z:, y) { g (.2:, y)in a domain D, then, denoting by yl (x)a solution of the first equation such that2/1(·T0) = I/os with (xm yo) €D» and byy2(x) a solution of the second equationwith the same initial data, we can provethat yl (sr) { y2 (.2:) for x Q xo in D. Butif in D we have the strict inequalityf(¤v, y) < s (¤v» y). then y1(¤¢)< y2 (rv) ferx>;1:0 in D and the curve y = y2 (0:) isa curve without contact.As an example let us consider the differ-ential equation (168). We have already

262 Differential Equations in Applicationsshown that if G > G0, the equation allowsfor two finite singular points, a saddlepoint and a nodal point, which are pointsof intersection of the straight line my —Gzx = 0 and the parabola 1 - py —{—qG2:c2 = O. The segments of the straight lineand parabola that link these two finite sin-gular points are curves without contact andthey specify a region of the plane which wedenote by A. If we putX (fm y) = 1 —- py + 062332,Y (rv, y) = 2y (my - 6%),it is easy to see that vector (X, Y) is di-rected on the boundary of A outwardexcept at the singular points. Hence, if therepresentative point emerges from any inte-rior point of A and moves along an integralcurve with t decreasing, it cannot leave Awithout passing through one of the singularpoints. But since inside A we haveX (as, y) < 0, the singular point that isattractive is the nodal point.Finding the slopes of the exceptionaldirections for the saddle point, we can seethat one of the exceptional straight linespasses through A. This implies that anintegral curve that is tangent to thisstraight line at the saddle point must enterA and then proceed to the nodal point.

Ch. 2. Qualitative Methods 2632.16 The Topographical Systemof Curves. The Contact CurveEarlier we noted that in the qualitativesolution of differential equations it is expe-dient to use curves without contact. It mustbe noted, however, that there is no generalmethod of building such curves that wouldbe applicable in every case. Hence, theimportance of various particular methodsand approaches in solving this problem.One approach is linked to the selection ofan appropriate topographical system ofcurves. Here a topographical system of curvesdehned by an equation cb (x, y) = C, withC a real-valued parameter, is understood tobe a family of nonintersecting, envelopingeach other, and continuously dijjerentiablesimple closed curves that completely jill adoubly connected domain G in the phaseplane.*li the topographical system is selectedin such a manner that each value of para~meter C is associated with a unique curve,then, assuming for the sake of deiinitenessthat the curve corresponding to a deiinitevalue of C envelopes all the curves withsmaller values of C (which means that the"size" oi the curves grows with C) and thatno singular points of the corresponding* Other definitions of a topographical system alsoexist in the mathematical literature, but essentiallythey differ little from the one given here.

264 Differential Equations in Applicationsdifferential system lie on the curves be-longing to the topographical system, wearrive at the following conclusion. If weconsider the function (D (cp (t), ip (t)) wherex = cp (t) and y = ip (t) are the parametricequations of the trajectories of the differen-tial system of the (122) type and calculatethe derivative with respect to t, that is,d<1>(<.¤(¢>» t\>(¢)) 0‘D(¤¤. y)‘—i?":—".w*X (‘”· ylthen the curves (cycles in our case) withoutcontact are the curves of the topographicalsystem on which the derivative d€D/dtis a function of fixed sign. lf, in addition,d€D/dt is positive on a certain curve be-longing to the topographical system, thensuch a curve, being a cycle without contact,possesses the property that all phase tra-jectories intersecting it leave, with thepassage of time i, the finite region boundedby it. And if d€D/dt is negative, all thephase trajectories enter this region. Thisimplies, for one thing, that if in an annularregion completely filled with curves belong-ing to the topographical system the deriv-ative d<D/dt is a function of fixed sign,the region cannot contain any closed tra-jectories, say, the limit cycles of the dif-ferential system. Limit cycles may existonly in the annular regions where the

Ch. 2. Qualitative Methods 265derivative d€D/dt is a function with alter-nating signs.To illustrate this reasoning, we considerthe differential systemd dgif-=y—·:v+w3. —H%·=—rv—y+y3, (184)which has only one singular point in thefinite part of the plane, O (0, O), a stablefocal point. For the topographical systemof curves we select the family of concentriccircles centered at point O (O, O), that is,the family of curves specified by the equa-tion x2 + y2 == C, with C a positive pa-rameter. In our case the derivative d€D/dtis given by the relationshipd<D-5 =—· — 2 (wz + y2) —l— 2 (¤¢" + 1%*)..-which in polar coordinates x = r cos 6 andy = r sin 6 assumes the form-S4?)-—..:. ——2r2-{— 2r’* (cost 9 —[—sin*= 9) = —2r2·\~2r’* ( cos 49).Noting that the greatest and smallest valuesof the expression in parentheses are 1 and1/2, respectively, we arrive a the_ followingconclusion. For r greater than I/2 the valueof the derivative dCD/dt is positive Wh11Gfor r less than unity it is negative. On thebasis of the Poincaré-Bendixson criterion18-0770

266 Differential Equations in Applicationswe conclude (if we replace t with —~t insystem (184)) that the annular region bound-ed by the circles x2 + y2 -:1 and x2 +y2 == 2 contains a limit cycle of system(184), and this limit cycle is unique.To prove the uniqueness it is sufficientto employ the Dulac criterion for a doublyconnected domain: if there exists a functionB (x, y) that is continuous together withits first partial derivatives and is such thatin adoubly connected domain G belongingto the domain of definition of system (122)the function6(BX) 0(BY)01: + Oyis of fixed sign, then in G there can be nomore than one simple closed curve consistingof trajectories of (122) and containing withinit the interior boundary of G.ln our case, selecting B (x, y) E 1 forsystem (184), we Hind that OX/0x +6Y/dy = 3 (x2 —i— y2) —— 2 and, as can easilybe seen, in the annular region with bound-aries x2 —|— y2 :1 and x2 —|— y2 = 2 theexpression 3 (x2 + y2) -— 2 always retainsits sign. Allowing now for the type of thesingular point O (O, O), we conclude thatthe cycle is an unstable limit cycle.To return to the problem oi buildingcurves without contact in the general case,let us examine a somewhat different way

Ch. 2. Qualitative Methods 267of using the topographical system ofcurves. This approach is based on thenotion of a contact curve. Introduction ofthis concept can be explained by the factthat if the derivative dd)/dt vanishes ona set of points of the phase plane, this setconstitutes the locus of points at which thetrajectories of the differential system touchthe curves belonging to the topographicalsystem. Indeed, the slope of the tangent toa trajectory of the differential system isY/X and the slope of the tangent to a curvebelongirg to the topographical system is—-(Gil 0:1:)/(GCD/Gy). Thus, when0 IJ 6iDthe slopes coincide, orL - -@/23X I- G1! , Gy °Hence, the locus of points at which thetrajectories of the differential system of the(122) type touch the curves belonging to thetopographical system (defined by the equa-tion (D (1, y) =-·· C) is called a contact curve.The equation of a contact curve has the(185) form. Of particular interest here isthe case where the topographical systemcan be selected in such a manner that eitherthe contact curve itself or a real branch ofthis curve proves to be a simple closedcurve. Then the topographical system con-

268 Differential Equations in Applications/ \x\ /@“\ /\ /Fig. 98tains the "greatest" and "smallest" curvesthat are touched either by a contact curveor by a real branch of this curve (in Fig-ure 98 the contact curve is depicted by adashed curve). If the derivative d€D/dtis nonpositive (nonnegative) on the "great-est" curve specified by the equationCD (sc, y) = C1 and nonnegative (nonposi-tive) on the "smallest" curve specified bythe equation (D (az, y) = C2, then the annu-lar region bounded by these curves containsat least one limit cycle of the differentialsystem studied.Specifically, in the last example (see(184)% the contact curve specified by theequation :1:2 -1- y2 =·- .154 -{- y4 is a closedcurve (Figure 99), which enables us to findthe °°gI‘eatest" and "smallest" curves in theselected topographical system that aretouched by the contact curve. Indeed, if

Ch. 2. Qualitative Methods 269HFig. 99we note that in polar coordinates the equa-tion of the contact curve has the form rz =(c0s‘ 9 + sin‘ 6)*, we can easily {ind theparametric representation of this curve:x __ cos 9 y __ sin 9l/c0s* 9—|—sin*9 ’ _ }/cos49—{—sin·*9Allowing now for the fact that the greatestand smallest values of r2 are, respectively,2 and l, we conclude that the "greatest”and "smal1est" curves of the topographical

270 Differential Equations in Applicationssystem :::2 —{— y2 == C are the circles :1:2 —}—y2 == 1 and :::2 + y2 == 2, respectively.These circles, as we already know, specify anannular region containing the limit cycleof system (184).2.17 The Divergence of a VectorField and Limit CyclesReturning once more to the general case,we note that in building the topographicalsystem of curves we can sometimes employthe right-hand side of system (122). Ithas been found that the differential systemmay be such that the equation0X GYwhere A is a real parameter, specifies thetopographical system of curves.For instance, if we turn to the differentialsystem (184), Eq. (186) assumes the form3 (x2 —|— y2) — 2 = it. Then, assuming thatA = (7t -[— 2)/3, with A 6 (-2, —|—oo), wearrive at the topographical system of curvesx2 + y2 == A used above.A remark is in order. If the "greatest" and"smallest" curves merge, that is, the contactcurve or a real branch of this curve coin-cides with one of the curves belonging tothe topographical system, such a curve isa trajectory of the differential system.

C11. 2. Qualitative Methods 271Let us consider, for example, the differ-ential system (173):3;%: —y+x (i——¤>2—y2>,gé/·= $+9 (1- $2-*92)-For this systemGX-5;+%;; 2**4 ($2+!!2),and Eq. (186) assumes the form 2 -4 (:1:2 + y2) :: X. If instead of parameter R.we introduce a new parameter A =(2 —- K)/4, with it E (-—<x>, 2), then thelatter equation, which assumes the formazz —|- y2 = A, defines the topographicalsystem of curves. The contact curve in thiscase is given by the equation (:1:** -{— y2) X(.2:2 -|— y2 —— 1) == 0 and, as we see, its realbranch :v“ + y2 == 1 coincides with one ofthe curves belonging to the topographicalsystem. This branch, as shown on pp. 237-239, proves to be a limit cycle of system(173).The last two examples, of course, illus-trate, a particular situation. At the sametime the very idea of building a topograph-ical system of curves by employing theconcept of divergence proves to be fruitfuland leads to results of a general nature.We will not dwell any further on these re-sults, but only note that the followingfact may serve as justification for what

272 Differential Equations in Applicationshas been said: if a diyferential system of the(122) type has a limit cycle L, there exists areal constant lt and a positive and continu-ously diyfferentiable in the phase plane func-tion B (sc, y) such that the equation0(BX) 0(BY) _—*a?·+*a;·` ·· Xspecifies a curve that has a finite real branchcoinciding with cycle L.Let us consider, for example, the differ-ential system-%;:.-.2:1;-· 2:1:3 —{— cozy- Bxyz —{·- y3,%—= ——¢v°+=v2y+¤¤y2.which has a limit cycle specified by theequation x2 —I— y2 = 1. For this system thedivergence of the vector field is0X GY··5Tz:·—_"‘··5!·;·T— 2*** 5$2—·· 3yz.We can easily verify that there is not asingle real lt for which the equation 2 —5.1:2 ·— 3y2 = lt specifies a trajectory of theinitial differential system. But if we takea function B (x, y) = 3:::2 —- 4xy + 7y2 +3, then0(BX) 0 (BY) ___ .*"a;—+_a;,‘—(“2+y“—‘)>< (-— 23x2-1-16xy-25y2-20)-14,

Ch. 2. Qualitative Methods 273and the equation_E°.Q.)Q+ ..21.iYL Z .. 14Ox Oyspecifies a curve whose finite real branch$2 + y2 -1 =·-· 0 proves to be the limitcycle.In conclusion we note that in studyingconcrete differential models it often provesexpedient to employ methods not discussedin this book. Everything depends on thecomplexity of the differential model, onhow deep the appropriate mathematicaltools are developed, and, of course, on theerudition and experience of the researcher.

Selected ReadmgsAmel’kin, V.V., and A.P. Sadovskii, MathematicalModels and Dijferential Equations (Minsk:Vysheishaya Shkola, 1982) (in Russian).Andronov, A.A., A.A. Vitt, S.E. Khaikin, andN.A. Zheleztsov, Theory of Oscillations (Read-ing, Mass.: Addison-Wesley, 1966).Braun, M., C.S. Cleman, and D.A. Drew (eds.),Dijferential Equation Models (New York:Springer, 1983).Derrick, W.R., and S.I. Grossman, ElementaryDijferential Equations with Applications, 2nded. (Reading, Mass.: Addison—Wesley, 1981).Erugin, N'.P., A Reader in the General Course ofDiyjerential Equations (Minsk: Nauka i Tekh-nika, 1979) (in Russian).Ponomarev, K.K., Constructing Dijierential Equa-tions (Minsk: Vysheishaya Shkola, 1973) (inRussian).Simmons, G.F., Diyfferential Equations with Ap-plications and Historical Notes (New York:McGraw-Hill, 1972).Spiegel, M.H., Applied Diyfferential Equations(Englewood Cliffs, NJ.: Prentice—I·Iall, 1981).

AppendicesAppendix 1.Derivatives of ElementaryFunctionsFunction I DerivativeC (constant) 0rc 1x" nx""11 _ 1.2: x21 zzF " Um?_. 1/ `_"".:'I J: 2 ]/J:... 11;/ x rz';/x”"1cx exax ax In aIn :1: -1-xlog x L log e= —·L-—° :2: ° rv ln a10 1 1 ~ 0.4343E10 x x 0g10 8 ~ Tsin x cos xcos :1: —- sin x

276 AppendicesFunction ‘ Derivative1tan :1: -52-: secz as00s :1:1 _ 2 _cot as ———r;——- - csc ILsm xsin xsecu: -——?=tanxseca:cos szc0s xCSCIB ···COt·Il7CSC.IJSID .1:. 1SIH"1 I1?II, 1 ___ x2COS .27 ······*····1/1- me1tan“1 x —————1~|-- x21colfl J:1—l~ :1:21 1SCG- Il? ***—··.:;*.;·—·IB |/ xg ~— 11csc*' .77:1: |/ :1:2- 1sinh sz cosh xcosh 1: sinh 1:n x ·——·—ta h 1c0sh2a:1C01}h ···+——x s1nh2:1:_ 1Sll”11l`1 :12 `7"`;“`—_"———-1/ 1 + $2

Appendices 277Function I Derivative(:0511-1 ‘/C , SU >'|/ xg- 1l:.mh”1 .2: ..i... _1-_—-:t2 1 I 'T I < 11-1 ....coth 2: 1__x2,I33]>i

Appendix 2.Basic IntcgralsPower-Law FunctionsIn xn+1S 22 d$=—"7z·-TI_·T· (TL =}$S -E1-£=1n I .2: |xTrigonomctric FunctionsSsin as dx: ·——c0s:vSc0s:vd;z:=sinxStanxdx= —ln | cosxlSc0txda:=1n l sinxldxS -7* = (-HI} {Bcos asS .;*.2;.. .. tsinza: — -_-C0 xFractional Rational Functionsda: _ 1 _ xS a=+x¤—7•T""“ ‘(1:)dx 1 _ :c$;¤::.;?;7“*“" 1 (7)::-2-2;-111-%;- (| :1: | <a)

Appendices 279dx 1 _ xSmw:7°°"h 1 (7)=—Q%1¤%;—% (I ¤= I >¤)Exponential FunctionsS ex exS ax dx: iIn aHypcrbolic FunctionsS sinh x dx: cosh xS cosh x dx: sinh xStanhx dx: ln lcoshxlScothxdx=ln I sinhxldxS = tanh {IJdxS 1: ···· coth SBIrrational Functionsdx x= · 11-1 ...)S l/az-3:2 sm ( ad . ————S =cosh’1(-ig-)=lH|(=¤-I-]/¤¤”··°2)}

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