Prepared ByAnnie ak Joseph

Prepared ByAnnie ak Joseph Session 2008/2009

KNF1023Engineering

Mathematics II

Introduction to ODEs

Learning Objectives

Apply an ODEs in real life application

Solve the problems of ODEs

Describe the concept of ODEs

Introduction to ODEs

Order ofODE

Introductionto ODEs

Solving anODE –general,particular,exactsolutions

Basic Concept

An ordinary differential equation is anequation with relationship betweendependent variable (“y”), independentvariable (“x”) and one or morederivative of with respect to .

Example:1.

2.

3.

y x

45, xy

8,, xyy

xyxyyyx ,,,,,, 31022

Basic Concept

Ordinary Differential equations differentfrom partial differential equations

Partial Differential equations-> unknownfunction depends on two or morevariables, so that they are morecomplicated

02

2

2

2

dy

Vd

dx

Vd

Order of ODEs:

The order of a differential equation is theorder of the highest derivative involvedin the equation.

Example:

1.2.3.4.

xy cos, 04,, yy

22,,,,,,2 )2(2 yxyeyyx x xyxyyyx ,,,,,, 31022

Arbitrary Constants

An arbitrary constant, often denoted by aletter at the beginning of the alphabetsuch as A, B,C, , etc. may assumevalues independently of the variablesinvolved. For example in , c1and c2 are arbitrary constants.

212 cxcxy

21 , cc

Solving of an Ordinary DifferentialEquations

A solution of a differential equation is arelation between the variables which isfree of derivatives and which satisfies thedifferential equation identically.

Solving of an Ordinary DifferentialEquations

Example 1:

06'' xy

Cxxdxdx

dyy 2, 36

DCxxdxCxy 32 )3(

Concept of General Solution

A solution containing a number ofindependent arbitrary constants equal tothe order of the differential equation iscalled the general solution of the equation.

We regard any function y(x) with Narbitrary constants in it to be a generalsolution of N th order ODE in y=y(x) if thefunction satisfies the ODE.

Concept of General Solution

Example 2 : is a solutionfor ODE

DCxxxy 38)(

xy 48''

xdx

ydy 48

2

2,,

Cxxdxy 2' 2448

DCxxdxCxy 32 8)24(

Particular Solution

When specific values are given to at leastone of these arbitrary constants, thesolution is called a particular solution.

Example 3:

Dxxxy 28)( 3

58)( 3 Cxxxy

158)( 3 xxxy

Exact Solution

A solution of an ODE is exact if thesolution can be expressed in terms ofelementary functions.

We regards a function as elementary if itsvalue can be calculated using an ordinaryscientific hand calculator.

Exact Solution

Thus the general solutionof the ODE is exact.

We may not able to find exact solutionfor some ODEs. As example, considerthe ODE

DCxxxy 38)(

xy 48''

dxx

xy

x

x

dx

dy

)sin(

)sin(

Applications of ODEs

Summary

Order of ODE

Solving an ODE

general, particular, exact solutions

ODEs

Prepared ByAnnie ak Joseph

Prepared ByAnnie ak Joseph Session 2008/2009