Ordinary Differential Equationsgn.dronacharya.info/.../Unit-3/initial-value-problem.pdfDEFINITION: initial value problem An initial value problem or IVP is a problem which consists

Ordinary Differential Equations

1.1 Definitions and Terminology

1.2 Initial-Value Problems

1.3 Differential Equation as Mathematical Models

CHAPTER 1 Introduction 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 or

With respect to is defined as

The derivative is also denoted by or


)(xfy y )(xf

x

h

xfhxf

dx

dy

h

)()(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 independent variables .

Differential Equations are classified by type, order and linearity.


Differential Equations are classified by type, order and linearity.

TYPE

There 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

2

2

2

2

2

x

v

y

u

ORDER

The 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

or

linear 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 y

nonlinear first-order ordinary differential equation

nonlinear function of y

nonlinear second-order ordinary differential equation

power not 1

nonlinear fourth-order ordinary differential equation


0)sin(2

2

ydx

yd

xeyyy 2)1( '

02

4

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 ODEs

DEFINITION: solution of an ODE

Any function , defined on an interval I and possessing at least n derivatives that are continuous

on 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 n

derivatives 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 curve

A 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 of

solutions 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


Example:


2' yy

xkex 2)(φ2' yy

xkex 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


Example:


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

Order Differential Equation-Second

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)=3

The 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 solution

A 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


Verify by implicit differentiation that the given equation implicitly defines a solution of the differential equation


Cyxxxyy 232 22

0)22(34 ' yyxxy


Verify 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(' n

n

n

yyyxfdx

yd

I

10)1(

10'

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

110 ,...,, nyyy

1.2 Initial-Value Problem THEOREM: Existence of a Unique Solution

Let R be a rectangular region in the xy-plane defined by that contains the point in its interior. If and are continuous on R, Then there exists some interval contained in and a unique function defined on that is a solution of the initial value problem.

dycbxa ,

),( 00 yx ),( yxf

yf /

0,: 000 hhxxhxI

bxa

)(xy0I

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Ordinary Differential Equationsgn.dronacharya.info/.../Unit-3/initial-value-problem.pdfDEFINITION: initial value problem An initial value problem or IVP is a problem which consists