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Ordinary Differential Equations
S.-Y. LeuSept. 21,28, 2005
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1.1 Definit ions and Terminology
1.2 Init ial-Value Problems 1.3 Differential Equation as Mathematical Models
CHAPTER 1Introduction to Differential Equations
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DEFINITION: differential equationAn equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a
differential equation (DE(.
)Zill, Definition 1.1, page 6(.
1.1 Definit ions and Terminology
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Recall CalculusDefinit ion of a DerivativeIf , the derivative of or
With respect to is defined as
The derivative is also denoted by or
1.1 Definit ions and Terminology
)(xfy = y )(xfx
h
xfhxf
dx
dyh
)()(lim0
−+=→
dx
dfy ,' )(' xf
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Recall the Exponential function
dependent variable: y independent variable: x
1.1 Definit ions and Terminology
xexfy 2)( ==
yedx
xde
dx
ed
dx
dy xxx
22)2()( 22
2
==
==
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Differential Equation: Equations that involve dependent variables and their derivatives with respect to the independentvariables.
Differential Equations are classified by
type, order and l inearity.
1.1 Definit ions and Terminology
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Differential Equations are classified bytype, order and l inearity.
TYPEThere are two main types of differential
equation: “ordinary” and “partial”.
1.1 Definit ions and Terminology
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Ordinary differential equation (ODE( Differential equations that involve only ONE independent variable are called ordinary differential equations.
Examples:
, , and
only ordinary (or total ( derivatives
1.1 Definit ions and Terminology
xeydx
dy =+ 5 062
2
=+− ydx
dy
dx
yd yxdt
dy
dt
dx +=+ 2
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Partial differential equation (PDE(Differential equations that involve two or more independent variables are called partial differential equations.Examples:
and
only partial derivatives
1.1 Definit ions and Terminology
t
u
t
u
x
u
∂∂−
∂∂=
∂∂
22
2
2
2
x
v
y
u
∂∂−=
∂∂
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ORDERThe order of a differential equation is the order of the highest derivative found in the DE.
second order f irst order
1.1 Definit ions and Terminology
xeydx
dy
dx
yd =−
+ 45
3
2
2
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1.1 Definit ions and Terminology
xeyxy =− 2'
3'' xy =
0),,( ' =yyxFfirst order
second order 0),,,( ''' =yyyxF
Written in differential form: 0),(),( =+ dyyxNdxyxM
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LINEAR or NONLINEAR
An n-th order differential equation is said to be l inear if the function
is linear in the variables
there are no multiplications among dependent variables and their derivatives. All coeff icients are functions of independent variables.
A nonlinear ODE is one that is not linear, i.e. does not have the above form.
1.1 Definit ions and Terminology
)1(' ,..., −nyyy
)()()(...)()( 011
1
1 xgyxadx
dyxa
dx
ydxa
dx
ydxa
n
n
nn
n
n =++++ −
−
−
0),......,,( )(' =nyyyxF
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LINEAR or NONLINEAR
orlinear first-order ordinary differential equation
linear second-order ordinary differential equation
linear third-order ordinary differential equation
1.1 Definit ions and Terminology
0)(4 =−+ xydx
dyx
02 ''' =+− yyy
04)( =+− xdydxxy
xeydx
dyx
dx
yd =−+ 533
3
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LINEAR or NONLINEAR
coefficient depends on ynonlinear first-order ordinary differential equation
nonlinear function of ynonlinear second-order ordinary differential equation
power not 1 nonlinear fourth-order ordinary differential equation
1.1 Definit ions and Terminology
0)sin(2
2
=+ ydx
yd
xeyyy =+− 2)1( '
024
4
=+ ydx
yd
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LINEAR or NONLINEAR
NOTE:
1.1 Definit ions and Terminology
...!7!5!3
)sin(753
+−+−= yyyyy ∞<<−∞ x
...!6!4!2
1)cos(642
+−+−= yyyy ∞<<−∞ x
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Solutions of ODEsDEFINITION: solution of an ODEAny function , defined on an interval I and possessing at least n derivatives that are continuouson I, which when substituted into an n-th order ODE reduces the equation to an identity, is said to be a
solution of the equation on the interval. )Zill, Definition 1.1, page 8(.
1.1 Definit ions and Terminology
φ
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Namely, a solution of an n-th order ODE is a function which possesses at least nderivatives and for which
for all x in I
We say that satisfies the differential equation on I.
1.1 Definit ions and Terminology
0))(),(),(,( )(' =xxxxF nφφφ
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Verification of a solution by substitution Example:
left hand side:
right-hand side: 0
The DE possesses the constant y=0 trivial solution
1.1 Definit ions and Terminology
xxxx exeyexey 2, ''' +=+=
xxeyyyy ==+− ;02 '''
0)(2)2(2 ''' =++−+=+− xxxxx xeexeexeyyy
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DEFINITION: solution curveA graph of the solution of an ODE is called a solution curve, or an integral curve of
the equation.
1.1 Definit ions and Terminology
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1.1 Definit ions and Terminology
DEFINITION: famil ies of solutions A solution containing an arbitrary constant (parameter) represents a set ofsolutions to an ODE called a one-parameter
family of solutions. A solution to an n−th order ODE is a n-parameter family of
solutions .
Since the parameter can be assigned an infinite number of values, an ODE can have an infinite number of solutions.
0),,( =cyxG
0),......,,( )(' =nyyyxF
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Verification of a solution by substitution Example:
1.1 Definit ions and Terminology
2' =+ yyxkex −+= 2)(φ
2' =+ yyxkex −+= 2)(φxkex −−=)(φ '
22)(φ)(φ ' =++−=+ −− xx kekexx
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Figure 1.1 Integral curves of y’+ y = 2 for k = 0, 3, –3, 6, and –6.
©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
xkey −+= 2
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Verification of a solution by substitution Example:
1.1 Definit ions and Terminology
1' +=x
yy
Cxxxx += )ln()(φ 0>x
Cxx ++= 1)ln()(φ '
1)(φ
1)ln(
)(φ ' +=++=x
x
x
Cxxxx
for all ,
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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure 1.2 Integral curves of y’ + ¹ y = e x
for c =0,5,20, -6, and –10. x
)(1
cexex
y xx +−=
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Second-Order Differential Equation
Example: is a solution of
By substitution:
)4sin(17)4cos(6)(φ xxx −=016'' =+ xy
0φ16φ
)4sin(272)4cos(96φ
)4cos(68)4sin(24φ
''
''
'
=++−=
−−=xx
xx
0),,,( ''' =yyyxF
( ) 0)(φ),(φ),(φ, ''' =xxxxFadmission.edhole.com
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Second-Order Differential EquationConsider the simple, l inear second-order equation
,
To determine C and K, we need two initial conditions, one specify a point lying on the solut ion curve and the
other its slope at that point, e.g. ,
WHY ???
012'' =− xy
xy 12'' = Cxxdxdxxyy +=== ∫∫ 2'' 612)('
KCxxdxCxdxxyy ++=+== ∫∫ 32' 2)6()(
Cy =)0('Ky =)0(
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Second-Order Differential Equation
IF only try x=x1, and x=x2
It cannot determine C and K,
xy 12'' =KCxxy ++= 32
KCxxxy
KCxxxy
++=
++=
23
22
13
11
2)(
2)(
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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure 2.1 Graphs of y = 2x³ + C x +K for various values of C and K.
e.g. X=0, y=k
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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure 2.2 Graphs of y = 2x³ + C x + 3 for various values of C.
To satisfy the I.C. y(0)=3The solution curve must pass through (0,3)
Many solut ion curves through (0,3)
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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Figure 2.3 Graph of y = 2x³ - x + 3.
To satisfy the I.C. y(0)=3,y’(0)=-1, the solution curve must pass through (0,3)having slope -1
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Solutions General Solution: Solutions obtained from integrating the differential equations are called general solutions. The general solution of a nth order ordinary differential equation contains n arbitrary constants resulting from
integrating times. Particular Solution: Particular solutions are the solutions obtained by assigning specific values to the
arbitrary constants in the general solutions. Singular Solutions: Solutions that can not be expressed by the general solutions are called singular
solutions.
1.1 Definit ions and Terminology
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DEFINITION: implicit solutionA relation is said to be an implicit solution of an ODE on an interval I provided there exists at least one function that satisfies the relation as well as the differential equation on I.
a relation or expression that defines a solution implicitly.
In contrast to an explicit solution
1.1 Definit ions and Terminology
0),( =yxG
)(xy φ=
φ
φ
0),( =yxG
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DEFINITION: implicit solutionVerify by implicit differentiation that the given equation implicitly defines a solution of the differential equation
1.1 Definit ions and Terminology
Cyxxxyy =−−−+ 232 22
0)22(34 ' =−++−− yyxxy
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DEFINITION: implicit solutionVerify by implicit differentiation that the given equation implicitly defines a solution of the differential equation
1.1 Definit ions and Terminology
Cyxxxyy =−−−+ 232 22
0)22(34 ' =−++−− yyxxy
0)22(34
02234
02342
/)(/)232(
'
'''
'''
22
=−++−−==>=−++−−==>=−−−++==>
=−−−+
yyxxy
yyyxyxy
yxxyyyy
dxCddxyxxxyyd
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Conditions Init ial Condition: Constrains that are specified at the initial point, generally time point, are called initial conditions. Problems with specified initial conditions
are called initial value problems.
Boundary Condition: Constrains that are specified at the boundary points, generally space points, are called boundary conditions. Problems with specified boundary conditions are called boundary value
problems.
1.1 Definit ions and Terminology
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First- and Second-Order IVPS Solve:
Subject to:
Solve:
Subject to:
1.2 Init ial-Value Problem
00 )( yxy =
),( yxfdx
dy =
),,( '2
2
yyxfdx
yd =
10'
00 )(,)( yxyyxy ==admission.edhole.com
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DEFINITION: init ial value problem An init ial value problem or IVP is a
problem which consists of an n-th order ordinary differential equation along with n initial conditions defined at a point found in the interval of definition differential equation initial conditions
where are known constants.
1.2 Init ial-Value Problem
0x
),...,,,( )1(' −= nn
n
yyyxfdx
yd
I
10)1(
10'
00 )(,...,)(,)( −− === nn yxyyxyyxy
110 ,...,, −nyyyadmission.edhole.com
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1.2 Init ial-Value ProblemTHEOREM: Existence of a Unique Solution
Let R be a rectangular region in the xy-plane defined by that containsthe point in its interior. If and are continuous on R, Then there
exists some interval contained in anda unique function defined on that is a solution of the initial value problem.
dycbxa ≤≤≤≤ ,
),( 00 yx ),( yxf
yf ∂∂ /
0,: 000 >+<<− hhxxhxI
bxa ≤≤)(xy
0I
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