Ordinary Differential Equations S.-Y. Leu Sept. 21,28, 2005

Ordinary Differential Equations

S.-Y. LeuSept. 21,28, 2005

1.1 Definitions and Terminology

1.2 Initial-Value Problems 1.3 Differential Equation as Mathematical Models

CHAPTER 1Introduction to Differential Equations

DEFINITION: differential equation

An 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.)


Recall Calculus

Definition of a Derivative

If , the derivative of orWith respect to is defined as

The derivative is also denoted by or


)(xfy y )(xfx

h

xfhxf

dx

dyh

)()(lim

0

dx

dfy ,' )(' xf

Recall the Exponential function

dependent variable: y independent variable: x


xexfy 2)(

yedx

xde

dx

ed

dx

dy xxx

22)2()( 22

2

Differential Equation: Equations that involve dependent

variables and their derivatives with respect to the

independentvariables.

Differential Equations are classified by

type, order and linearity.


Differential Equations are classified by

type, order and linearity.

TYPEThere are two main types of differential equation: “ordinary”

and “partial .”


Ordinary differential equation (ODE)

Differential equations that involve only ONE independent variable are called ordinary differential equations.

Examples:

, , and

only ordinary (or total ) derivatives


xeydx

dy5 06

2

2

ydx

dy

dx

yd yxdt

dy

dt

dx 2

Partial differential equation (PDE)

Differential equations that involve two or more independent variables are

called partial differential equations.Examples:

and

only partial derivatives


t

u

t

u

x

u

22

2

2

2

x

v

y

u

ORDERThe order of a differential equation is the order of the highest derivative found in the DE.

second order first order


xeydx

dy

dx

yd

45

3

2

2


xeyxy 2'

3'' xy

0),,( ' yyxFfirst order

second order

0),,,( ''' yyyxF

Written in differential form: 0),(),( dyyxNdxyxM

LINEAR or NONLINEAR

An n-th order differential equation is said to be linear if the function

is linear in the variables

there are no multiplications among dependent variables and their derivatives. All coefficients are functions of independent variables.

A nonlinear ODE is one that is not linear, i.e. does not have the above form.


)1(' ,..., nyyy

)()()(...)()( 011

1

1 xgyxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n

0),......,,( )(' nyyyxF

LINEAR or NONLINEAR

orlinear first-order ordinary differential equation

linear second-order ordinary differential equation

linear third-order ordinary differential equation


0)(4 xydx

dyx

02 ''' yyy

04)( xdydxxy

xeydx

dyx

dx

yd 53

3

3

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


0)sin(2

2

ydx

yd

xeyyy 2)1( '

024

4

ydx

yd

LINEAR or NONLINEAR

NOTE:


...!7!5!3

)sin(753

yyy

yy x

...!6!4!2

1)cos(642

yyy

y x

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.(


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.


0))(),(),(,( )(' xxxxF n

Verification of a solution by substitution Example:

left hand side:

right-hand side: 0

The DE possesses the constant y=0 trivial solution


xxxx exeyexey 2, '''

xxeyyyy ;02 '''

0)(2)2(2 ''' xxxxx xeexeexeyyy

DEFINITION: solution curveA graph of the solution of an ODE is called a solution curve, or an integral curve

of the equation.



DEFINITION: families 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



2' yyxkex 2)(φ

2' yyxkex 2)(φxkex )(φ '

22)(φ)(φ ' xx kekexx

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



1' x

yy

Cxxxx )ln()(φ 0x

Cxx 1)ln()(φ '

1)(φ

1)ln(

)(φ '

x

x

x

Cxxxx

for all ,


Figure 1.2 Integral curves of y’ + ¹ y = ex

for c =0,5,20, -6, and –10. x

)(1

cexex

y xx

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, ''' xxxxF


Consider the simple, linear second-order equation

,

To determine C and K, we need two initial conditions, one specify a point lying on the solution 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(


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)(


Figure 2.1 Graphs of y = 2x³ + C x +K for various values of C and K.

e.g. X=0, y=k


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 solution curves through (0,3)


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

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.


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


0),( yxG

)(xy

0),( yxG

DEFINITION: implicit solutionVerify by implicit differentiation that the given equation implicitly defines a solution of the differential equation


Cyxxxyy 232 22

0)22(34 ' yyxxy

DEFINITION: implicit solutionVerify by implicit differentiation that the given equation implicitly defines a solution of the differential equation


Cyxxxyy 232 22

0)22(34 ' yyxxy

0)22(34

02234

02342

/)(/)232(

'

'''

'''

22

yyxxy

yyyxyxy

yxxyyyy

dxCddxyxxxyyd

Conditions Initial 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.


First- and Second-Order IVPS Solve:

Subject to:

Solve:

Subject to:

1.2 Initial-Value Problem

00 )( yxy

),( yxfdx

dy

),,( '2

2

yyxfdx

yd

10'

00 )(,)( yxyyxy

DEFINITION: initial value problem

An initial 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 Initial-Value Problem

0x

),...,,,( )1(' nn

n

yyyxfdx

yd

I

10)1(

10'

00 )(,...,)(,)( nn yxyyxyyxy

110 ,...,, nyyy

1.2 Initial-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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Ordinary Differential Equations S.-Y. Leu Sept. 21,28, 2005