Introduction to Differential Introduction to Differential EquationsEquations

CSE:108

Bappa SarkarLecturerCSE, IU, Kushtia

Definition:

A differential equation is an equation containing an unknown function and its derivatives.

32 xdxdy

032

2

aydxdy

dxyd

364

3

3

ydxdy

dxyd

Examples:.

y is dependent variable and x is independent variable, and these are ordinary differential equations

1.

2.

3.

Ordinary differential equations

Partial Differential EquationExamples:

02

2

2

2

yu

xu

04

4

4

4

tu

xu

tu

tu

xu

2

2

2

2

u is dependent variable and x and y are independent variables, and is partial differential equation.

u is dependent variable and x and t are independent variables

1.

2.

3.

Order of Differential Equation The order of the differential equation is order of the highest derivative in the differential equation.

Differential Equation ORDER

32 x

dxdy

0932

2

ydxdy

dxyd

364

3

3

ydxdy

dxyd

1

2

3

Degree of Differential Equation

Differential Equation Degree

032

2

aydxdy

dxyd

364

3

3

ydxdy

dxyd

0353

2

2

dxdy

dxyd

1

1

3

The degree of a differential equation is power of the highest order derivative term in the differential equation.

Linear Differential Equation A differential equation is linear, if

1. Dependent variable and its derivatives are of degree one,2. Coefficients of a term does not depend upon dependent

variable.

Example:

364

3

3

ydxdy

dxyd

is non - linear because in 2nd term is not of degree one.

.0932

2

ydxdy

dxydExample:

is linear.

1.

2.

Example:3

2

22 x

dxdyy

dxydx

is non - linear because in 2nd term coefficient depends on y.

3.

Example:

is non - linear because

ydxdy sin

!3

sin3yyy is non – linear

4.

First Order Ordinary Differential equation

8

Second order Ordinary Differential Equation

9

nth – order linear differential equation

1. nth – order linear differential equation with constant coefficients.

xgyadxdya

dxyda

dxyda

dxyda n

n

nn

n

n

012

2

21

1

1 ....

2. nth – order linear differential equation with variable coefficients

xgyxadxdyxa

dxydxa

dxydxa

dxdyxa n

n

nn

012

2

2

1

1 ......

10