APPENDIX

OrdinaryDifferentialEquations

PDE models are frequ ently solved by reducing them to one or more ODEs.This appendix contains a bri ef review of how to solve som e of the basicODEs that are encountered in this book. At the end of th e appendix areseveral exercises that should be solved by hand; the reader might want tocheck the solutions using a compute r algebra package .

For notation, we let y = y (x) be the unknown func tion . Derivatives willbe denoted by pr imes , i.e ., y ' = y'(X), y" = y"(X). Sometimes we use thedifferential notation y' = '!t .If f is a function, an antideriuatiue is definedas a fun ction F whose derivative is i, i.e., F'(x) = fex) . Antiderivativesare unique onl y up to an additive constant, and they are often denotedby the usual indefinite integral sign:

F(x) = f j(x )dx + C.

An arbitrary constant of integration C is added to the right side. How­eve r, in this last expression, it is sometimes impossible to evaluate th eantiderivat ive F at a particular value of x . For example, if [(x) = sin x /x ,then there is no simple formula for the antiderivative; that is,

f sinxF(x ) = -- dx

x

cannot be expressed in closed form in terms of eleme ntary fun ctions, andthus we could not find , for example, F(Z). Th erefore, it is better to denoteth e antiderivative by an integral with a variable upp er limit,

F(x ) = f'f(s )ds + C,

"

198 A. Ordinary Differential Equations

where a is any constant (observe that a and C are not independent, sincechanging one changes the other) . By the fundamental theorem of calcu­lus , F '(x) = f(x)' Now, for example , the ant iderivative of sin x /x can bewritten (taking C = 0)

1x sin s

F(x) = -ds,o s

and easily we find that

12 sin sF(Z) = - ds ~ 1.605.

o s

First-Order EquationsAn ODE of the first order is an equation of the form

G(x , y, y/) = O.

There are three types of these equations that occur regularly in PDEs:separable, linear, and Bernoulli . The general solution involves an arbi­trary constant C that can be determined by an initial condition of theform y(xo) = Yo .

Separable Equations

A first-order equation is separable if it can be written in the form

dydx = f(x)g(y) .

In this case we separate variables to write

dy = f(x)dx .g(y )

Then we can integrate both sides to get

! dy = !f(X)dx + C,g(y)

which defines the solution implicitly. As noted above, sometimes the anti­derivatives should be written as definite integrals with a variable upperlimit of integration.

The simplest separable equation is the growth-decay equation

y/ = )"y ,

which has general solution

y = ce":

A . Ordinary Differential Equations 199

The solution mod els exponentia l growth if A > 0 and expone ntial decayin < o.

Linear Equations

A first-order linear equa tion is one of the form

.11' + p(x )y = q(x ).

This can be solved by multiplying through by an int egrating factor of theform

e.Cp (s)ds .

This turns the left side of the equa tion into a total derivative, and itbecomes

~ ( .11 expel ' P(S)dS)) = q(x) exp(l x

p(s)ds)

Now, both sides can be integrated from (l to x to find y . We illustra te th isprocedure with an example.E XAMP LE

Find an expression for the solutio n to the initial value problem

.11' + 2xy = JX, .11( 0) = 3.

The integrating factor is exp(j~Y 2sds) = exp(x2) . Multip lying bo th sidesof the equat ion by the integrati ng factor gives

(ye")' = JXe x: .

Now, integrating from () to X (while changing the dummy var iable ofintegration to s) gives

Solving for .11 gives

y(x) = e- X'(3 + r JSe s'ds) = 3e- x' + r JSes 2-

x' ds.10 10As is freque ntly the case, the integrals in this example cannot be com­puted eas ily, if at all, and we mu st write the solution in term s of integralswith variable limi ts. D

Bernoulli Equations

Bernoulli equations are nonlinear equations having the form

.11' + p(x)y = q(x)yn.

200 A. Ordinary Differential Equations

The transformation of dependent variables w = yl-n turns a Bernoulliequation into a first-order linear equation for w.

Second-Order EquationsSpecial Equations

Some second-order equations can be immediately reduced to a first­order equation. For example, if the equation has the form

G(x, y', y") = 0,

where y is missing, then the substitution v = y' reduces the equation tothe first-order equation

G(x , v , v') = 0.

If the second-order equation does not depend explicitly on the indepen­dent variable x, that is, it has the form

G(y, y', y") = 0,

then we again define v = y' . Then

" d,y =-ydx

So the equation becomes

dvdx

dv dy

dy dxdvdy v.

dvG(y , v , dy v) = 0,

which is a first-order equation in v = v(y).

Linear, Constant-Coefficient Equations

The equation

ay" + by' + cy = 0,

where a, b, and c are constants, occurs frequently in applications. Werecall that the general solution of a linear, second-order, homogeneousequation is a linear combination of two independent solutions. That is, ifYl(X) and yz(x) are independent solutions, then the general solution is

y = ClYl(X) + czYz(x),

where Cl and Cz are arbitrary constants. If we try a solution of the formy = emx , where m is to be determined, then substitution into the equationgives the so-called characteristic equation

am' + bm + c = °

A. Ordinary Differential Equations 201

for m . This is a quadratic polynomial that will have two roots, m j and mz.Three poss ibiliti es can occur: un equal real roots , equal real roots, andcomplex roots (whi ch mu st be complex conjugates).

Case (I). m j, mz rea l and unequal. In this case two independentsolutions are el111X and e111 2X

•

Case (II) . m j , mz real and equal, i.e., m ] = mz == m . In this case twoind ependent solutions are el11X and xel11X

•

Case (III ) . m j = a + i f3, mz = a - if3 are complex conjugate roots. Inthis case two real, ind ependent solutions are e" sin f3x and e" cos f3x .

Of particular imp ortance are the two equations y " + aZy = 0, wh ichhas gene ral soluti on y = C l cos ax + C2 sin ax, and y" - aZy = 0,which has general solution y = C] e- ax + czeax , or equivalently, y =C1 cosh ax + Cz sinh ax . These two equations occur so freque ntly that it isbest to memorize their solutions.

Cauchy-Euler Equations

It is difficult to solve second-order linear equations with variable coeffi­cients. Often , the reader may recall, power seri es methods are applied .Howeve r, there is a special equation that can be solved with simpleformulae, na mely , a Cauchy-Euler equation of the form

ax2y" + bxy' + cy = O.

Th is equation adm its power functions as solutions . Hen ce, if we try asolution of the form y = x'" , where m is to be determined , then we obtainupon substitution the characteristic equation

am(m - 1) + bm + c = O.

This quadratic equation has two roots, m j and mz. Thu s, there are threecases:

Case (I). m ], mz real and unequal. In th is case two independe ntsolutions are X1111 and X 1111

•

Case (II) . m j , mz real and equa l, i.e. , m ] = mz == m. In th is case twoindepende nt solutions are x l11 and x l11 In x .

Case (III ). m j = a + i{3, m2 = a - i{3 are complex conjugate roots . In th iscase two real , independ ent solut ions are XU sin( f3 ln x) and xa cos(f3 ln x) .

Particular SolutionsTh e gene ral solution of the inhomogeneous ODE

y" + p( x )y ' + q(x )y = [(x)

is

202 A. Ordinary Differential Equ ations

where Yl and yz are independent solutions of the homogeneous equat ion[when [(x) == 0], and yp is any particular solution to the inhomogeneousequation. For constant-coefficient equations a particular solutio n cansometimes be "guessed" from the form of [(x); the reader may recall thatthis guessing method is called the method of undetermined coefficients.In any case, however, there is a gen eral formula , called the variation o[pa­rameters formu la , which gives the particular solutio n in terms of the twolinearly independent solutions Yl and yz. The formula, which is derivedin elementary texts, is given by

There are several int rodu ctory texts on differential equations [see,for example, Boyce and DiPrim a (1995) or Edwards and Penney (2004)].Birkhoff and Rota (1978) and Hirsch , Smale, and Devaney (2004) are twomore advanced texts.

ExercisesSolve the following differential equations.

1. y' + 2y = e- x .

2. y' = - 3y .

3 . y" + By = O.

4. y' - xy = XZy z .

5. x2y" - 3xy' + 4y = O.

6 . y" + xy'Z = o.

7. y" + y' +Y = O.

8. yy" - y, 3 = O.

9. 2x2y" + 3xy ' - Y = O.

10. y" - 3y' - 4y = 2 sinx.

11 . y" + 4y = x sin 2x .

12. y' - 2xy = 1.

13. y" + 5y' + 6y = O.

, x2

14. Y = 1+3y3 '

15. y" - 6y = O.

Exercises

TABLE OF LAPLACE TRANSFORMS

u(t)

t"

sin at

cos at

sinh at

cosh at

H(t - a)u(t - a)

1 - erf ( -..E-)J4t

U( s)

---Ls-a

sa'+s'

sS2_a2

U(s - a)

20 3

(u * v)(t)

tu(t)

u(t) / t

u(at)

o(t - a)

U(s )V(s)

- U' (s)

!,OO U(r)dr

U(s / a) / a

References

I. M. Abra mowitz and l.A. Steg un , eds., Handbook of Mathematical Functions, DoverPubli cations, New York (1965) .

2. R.B. Bird , W E. Stewart , and E.N. Light foot, 'Transport Phenomena, John Wiley and Son s,New York (1960).

3. G. Birk ho ff and .l.C. Rota, OrdinunJ Differential Equations , John Wiley and Sons, NewYork (1978) .

4. W E. Boyce and R.C. DiPrima , Elementaru Diiiereruial Equations, 5th ed., John Wileyand Sons , New York (1995).

5. N.F. Britton , Essent ial Mathematical Biology, Springe r-Verlag, New York (2003).

6. H .S. Ca rslaw and .l.C . Jaeger, Conduction of Heat in Solids, Znd ed ., Clarendo n Press,Oxfo rd (1959).

7. A.J . Cho rin and J .E. Mars de n , A Mathematical ll1tmduction to Fluid Mechanics, 3rd ed .,Spri nger-Verl ag, New Yo rk (1993 l.

8. R.v. Churc hill, Complex Variables and Applications , 2nd ed. , McGraw-Hill , New York(1960)

9. R.Y. Churc hill, Operational Mathclnatics , 3rd ed .. McGraw-Hill , New York (1972).

10. L. Edel stein-Kosh er . Mathematicul Models in Biology , McGraw -Hill, New York (1988) .

11 C. H . Edwa rds and D. E. Penn ey . Differential Equations , 3rd ed , Pear son Educa tion ,Uppe r Saddle River , N,) (2004) .

12 1'. Grindrod , Pattern » u /lcl Wal'e,) , Cla rendon Press, Oxford (1991).

13. C.W Groetsch, l/l l'crse Problem: in the Mathematical Sciences, Vieweg, Braun-sch wcig/wiesbadcu (1993).

14. R.B. Guent he r and .J W. Le«, Partiul Differential Equations of Mathematical Physics andIntegral Equations, Dover Publi cations, New York (1992).

15. M.W Hirsch, S. Sma le . and R.L. Devan ey , Differential Equations , Dynamical Systems ,and an Introduction to Chao«. 2nd ed, Elsev ier (Academ ic Press) , San Diego (2004) .

206 References

16. M. Kot, Elements of Math ematical Ecology, Cambridge University Press, Cambridge(2001) .

17. F. John, Partial Differential Equation s, 4th ed., Springer-Verlag, New York (1982) .

18. C.C. Lin and L.A Segel, Mathematics Applied to Deterministic Models in the NaturalSciences, SIAM, Philadelphia (1989).

19. J .D. Logan , Introdu ction to Nonlinear Partial Differentia l Equations, Wiley-Int erscien ce,New York (1994).

20. J .D. Logan , Applied Math ematic s: Second edition , Wiley-lnterscien ce, New York (1997).

21. J .D. Logan , 'Transport Modeling in Hydrogeochemical Systems, Springer-Ver lag, New York(2001).

22. G. de Marsily, Quantitative Hydrogeology, Academic Press, New York (1987).

23. K.W. Morton and D.F. Mayers, Numerical Solution to Partial Differential Equations,Cambridge Univ. Press (1994).

24. J.D . Murray, Math emati cal Biology, Vol. 11, Spring er-Verlag, New York (2003).

25. M. Renardy and R.C. Rogers, An Introdu ction to Partial Differential Equations, Springer ­Verlag (1993).

26. L.A Segel, Math ematics Applied to Continuum Mechanics, Dover Publications, New York(1987) .

27. J. Smoller, Shock Waves and Reaction-DiffUsion Equations, 2nd ed., Spring er-Verlag(1995) .

28. W. Strauss, Introduction to Partial Differential Equations, John Wiley and Sons, New York(1992) .

29. R. Strichartz, A Guide to Distribution Theory and Fourier 'Transform s, CRC Press , BocaRaton (1994).

30. AN . Tychono v and AA. Samarskii, Equations of Mathematical Physics, Dover Public a­tions, New York (1990) .

31. G.B. Whitham, Linear and Nonlinear Waves, Wiley-lnterscience, New York (1974).

acoust ics , 35-38acoustic approximation , 37Airy functio n , 92 , 120age-struc ture , 172age-struc ture , stable, 175aggregation of cells, 191antide rivative , 197ato m , computationa l, 163

Bernoulli equation, 199Bessel's inequality, 104Bessel 's different ial equa tion , 149Bessel func tions, 150bio logical invasion , 30bou ndary condition , 3, 4boundary value problem , 4Burgers ' equation, 185

canonical form , 55Cau chy probl em, 58, 62Cauchy-Euler equa tion , 201Cauchy-Schwartz in equ ality, 108cha rac te ristics, 14, 66cha rac te ristic coordina tes , 13-1 4,

55cha rac te ristic parallelogram , 68complete, 105

Index

conse rvat ion law, 10, 11 , 32cons titutive relation , 11convection , 12conve rgence in L2 , 102conve rgence in mean square,

102convergence, pointwise, 102conve rgence, uniform , 102convolution , 84, 88Courant-Friedrichs-Levy (CFL)

condition , 166

d 'Ale rnbert 's formula, 65decay , 12diffusion , 16diffusion constant, 17diffusion equation, 17, 44diffusivity , 18dimen sionl ess vari able, 59Dirichlet condition , 19,45 ,129Dirichlet problem , 4Dirichlet principle, 46 , 141dispersion relat ion , 9divergen ce ope rator, 43divergence th eor em , 43, 140domain of dependence, 66Duham el 's principle, 77, 153

208

eigenfunction, 115eigenvalue, 115elliptic, 53energy argument, 118energy integral, 36, 141energy spectrum, 105, 109epidemic wave, 184error, mean square, 102error, pointwise, 101Euler's method, 162Euler-Lotka equation, 175

Fick's law, 17, 27finite difference method, 161Fisher's equation, 24flux, 10, 25Fourier coefficients 103, 107Fourier heat law, 18, 44Fourier integral theorem, 87Fourier series, 99, 103, 107Fourier transform, 86frequency spectrum, 108, 109fundamental solution, 27, 29, 62

Gauss-Seidel method, 168Gram-Schmidt orthogonalzation,

106Green's function , 29, 63Green's identities, 46, 140, 141

Haar wavelets , 107heat equation, 17,44heat kernel, 62Hermite polynomials, 106hyperbolic equation, 53

initial boundary value problem, 4initial condition, 4inverse problem, 156

Korteweg-de Vries equation,linearized, 92

L2[a, bJ, 101Laplace transform, 81Laplace transforms, table of, 203Laplace 's equation, 47, 136

Index

Laplacian, 44Laplacian in other coordinate

systems, 49Legendre polynomials, 106, 147Legendre's differential equation,

147Leibniz 's rule, 8light cone , 66linearity, 4logistics equation, 23

Malthus' law, 2, 23maternity function, 173maximum principle, 49Maxwell's equations, 52McKendrick-von Foerster

equation, 174moments, method of, 177

Neumann condition, 45nonlinear, 4nonlocal boundary condition, 174norm, 100nutrient absorption, 16

order of an equation, 4orthogonal, 99orthonormal, 101, 103

parabolic equation, 53Parseval's equality, 105, 109periodic boundary condition, 135,

137periodic function , 110perturbation equation, 188pi theorem, 59piecewise continuous, IIIpiecewise smooth, IIIplane wave , 8Poisson's equation, 45Poisson's integral formula, 139Poisson's integral representation,

63predator-prey model, 177principal part of an operator, 53

quantum mechanics, 39

Index

radiation condition , 19, 129reaction-advection equation , 14region of influen ce, 66ren ewal equ ation , 176Robin condition , 19, 129

sca lar product , 100Schrodinger equat ion, 40Schwart z class, 87separation of variables, 121solution to a POE, 5sound speed , 37square-integrable, 99stability , 164, 189standing wave , 38Sturm-Liouville problem , 11 4

209

support of a fun ction , 65

traffic flow, 16temperature of earth, 76tra veling wave 35, 181- 187TUring, Alan , 187

variation of param eters formula(ODE), 77, 202

variation of parameters formu la(PDE),79

vibrations of a string , 32

wave-front, 182wave equation , 35, 53wave speed, 35well-posed, 70

Undergraduate Texts in Mathematics

(continued fro m page iii

Franklin : Methodsof MathematicalEconomics.

Frazier: An Introduction to WaveletsThrough Linear Algebra

Gamelin: Complex Analysis.Gordon: Discrete Probability.Hairer/Wanner: Analysis by Its History.

Readings in Mathematics.Halmos: Finite-Dimensional Vector

Spaces. Second edition.Halmos: Naive Set Theory.Hammerlin/Hoffmann : Numerical

Mathematics.Readings ill Mathemati cs.

Harris/Hirst /Mossinghoff:Combinatorics and Graph Theory.

Hartshorne: Geometry: EuclidandBeyond.

Hijab: Introduction to Calculus andClassical Analysis.

Hilton/Holton/Pedersen: MathematicalReflect ions: In a Room with ManyMirrors.

Hilton/Holton/Pedersen: MathematicalVistas: From a Roomwith ManyWindows.

Iooss/Joseph: Elementary Stabilityand Bifurcation Theory. Secondedition.

Irving: Integers, Polynomials, and RingsA Course in Algebra

Isaac: The Pleasures of Probability.Readings in Mathemati cs.

James: Topological and UniformSpaces.

Janich: Linear Algebra.Janich: Topology.Janich: Vector Analysis.Kemeny/Snell: Finite MarkovChains.Kinsey: Topologyof Surfaces.KIambauer: Aspectsof Calculus.Lang: A First Course in Calculus.

Fifth edition.Lang : Calculus of Several Variables.

Third edition.Lang : Introduction to Linear Algebra.

Second edition.

Lang: LinearAlgebra. Third edition.Lang: Short Calculus: The Original

Edition of "A First Course inCalculus."

Lang: Undergraduate Algebra. Secondedition.

Lang: Undergraduate Analysis.Laubenbacher/Pengelley: Mathematical

Expeditions.Lax/Burstein /Lax: Calculus with

Applications and Computing.Volume I.

LeCuyer: College Mathematics withAPL.

LidI/Pilz: AppliedAbstract Algebra.Second edition.

Logan: AppliedPartial DifferentialEquations, Second edition.

Lovasz/Pelikan/Vesztergombi: DiscreteMathematics.

Macki-Strauss : Introduction to OptimalControl Theory.

Malitz: Introduction to MathematicalLogic.

Marsden /Weinstein: Calculus I, II, III.Second edition.

Martin : Counting: The Art ofEnumerative Combinatorics.

Martin: The Foundations of Geometryand the Non-Euclidean Plane.

Martin: Geometric Constructions.Martin: Transformation Geometry: An

Introduction to Symmetry.Millman/Parker: Geometry: A Metric

Approach with Models. Secondedition.

Moschovakis: Notes on Set Theory.Owen: A First Course in the

Mathematical Foundations ofThermodynamics.

Palka: An Introduction to ComplexFunction Theory.

Pedrick: A First Course in Analysis.Peressini/Sullivan/Uhl: The Mathematics

of Nonlinear Programming.

Undergraduate Texts in Mathematics

PrenowitzlJantosciak: Join Geometries.Priestley: Calculus : A Liberal Art.

Second edition .Protter/Morrey: A First Course in Real

Analysis. Second edition.Protter/Morrey: Intermediate Calculus.

Second edition .Pugh: Real Mathematical Analysis.Roman : An Introduction to Coding and

Information Theory.Ross: Elementary Analysis: The Theory

of Calculus .Samuel: Projective Geometry.

Readings in Mathematics.Saxe: Beginning Functional AnalysisScharlau/Opolka: From Fermat to

Minkowski .Schiff: The Laplace Transform : Theory

and Applications.Sethuraman: Rings , Fields , and Vector

Spaces : An Approach to GeometricConstructab ility.

Sigler: Algebra .SilvermanlTate: Rational Points on

Elliptic Curves .Simmonds: A Brief on Tensor Analysis.

Second edition.Singer: Geometry: Plane and Fancy.

Singer/Thorpe: Lecture Notes onElementary Topology andGeometry.

Smith: Linear Algebra . Third edition .Smith: Primer of Modern Analysis.

Second edition.Stanton/White: Constructive

Combinatorics.Stillwell: Elements of Algebra: Geometry ,

Numbers, Equations .Stillwell: Elements of Number Theory .Stillwell: Mathematics and Its History .

Second edition .Stillwell: Numbers and Geometry.

Readings in Mathematics.Strayer: Linear Programming and Its

Application s.Toth: Glimpses of Algebra and Geometry.

Second Edition.Readings in Mathematics.

Troutman: Variational Calculu s andOptimal Control. Second edition.

Valenza: Linear Algebra: An Introductionto Abstract Mathematics .

Whyburn/Duda: Dynamic Topolog y.Wilson: Much Ado About Calculus .