DIFFERENTIAL EQUATION

PRESENTED BY :Irfan Ali

Definition

A Differential Equation is an equation containing the derivative of one or more dependent variables with respect to one or more independent variables.

For example,

Language Of The Differential Equation:

DEGREE OF ODE ORDER OF ODE SOLUTIONS OF ODE

GENERAL SOLUTION PARTICULAR SOLUTION TRIVIAL SOLUTION SINGULAR SOLUTION EXPLICIT AND IMPLICIT SOLUTION

HOMOGENEOUS EQUATIONS NON-HOMOGENEOUS EQUTIONS INTEGRATING FACTOR

Classification of Differential Equation:

Type: Ordinary Partial

Order : 1st, 2nd, 3rd,....,nth

Linearity : Linear Non-Linear

Classifiation by Type:Ordinary Differential Equation

If a Differential Equations contains only ordinary derivatives of one or more dependent variables with respect to a single independent variables, it is said to be an Ordinary Differential Equation or (ODE) for short. For Example,

Partial Differential Equation

If a Differential Equations contains partial derivatives of one or more dependent variables of two or more independent variables, it is said to be a Partial Differential Equation or (PDE) for short.

For Example,

The order of the differential equation (either ODE or PDE) is the order of the highest derivative in the equation.

For Example,

Order = 3

Order = 2

Order = 1

Classifiation by Order :

Classifiation by Lineraity:

Linear

In other words, it has the following general form:

)()()()( 2,nfor and

)()()( 1,nfor now

)()()()(......)(a)(a

012

2

2

01

012

2

21

1

1nn

xgyxadxdyxa

dxydxa

xgyxadxdy

xa

xgyxadxdyxa

dxydxa

dxydx

dxydx n

n

n

n

y ......., ,y ,yin linear is 0),......,,,,F(iflinear be tosaid is ODEOrder n The

n21

)(th nyyyyx

NON-LINEAR :

• A nonlinear ODE is simply one that is not linear. It contains nonlinear functions of one of the dependent variable or its derivatives such as:

• siny ey ln y• Trignometric Exponential

Logarithmic • Functions Functions

Functions

Linear

For Example,

Likewise,

Linear 2nd Order ODE is

Linear 3rd Order ODE is

Non-Linear

For Example,

ODEOrderlinearxyyxyxxy

dyxdxxy

1 are which 505

0 5

st

xeyyxy

xyyxy

5

25 2

0

0cos51

2)4(

yy

yyeyyy x

FOR EXPMALE

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