Introduction to Differential Equations...ordinary differential equations Partial Differential Equation Examples: 0 2 2 2 2 w w w y u x 0 4 4 4 4 w w w t u x t u t u x u w w w w w w

Introduction to Differential

Equations

Mr. Raut S. R. Mr. K.S.K.College Beed

Definition:

A differential equation is an equation containing an unknown function

and its derivatives.

32 xdx

dy

032

2

aydx

dy

dx

yd

36

4

3

3

y

dx

dy

dx

yd

Examples:.

y is dependent variable and x is independent variable,

and these are ordinary differential equations

1.

2.

3.

ordinary differential equations

Partial Differential Equation

Examples:

02

2

2

2

y

u

x

u

04

4

4

4

t

u

x

u

t

u

t

u

x

u

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 xdx

dy

0932

2

ydx

dy

dx

yd

36

4

3

3

y

dx

dy

dx

yd

1

2

3

Degree of Differential Equation

Differential Equation Degree

03

2

2

aydx

dy

dx

yd

36

4

3

3

y

dx

dy

dx

yd

03

53

2

2

dx

dy

dx

yd

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:

36

4

3

3

y

dx

dy

dx

yd

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

.0932

2

ydx

dy

dx

ydExample:

is linear.

1.

2.

Example: 3

2

22 x

dx

dyy

dx

ydx

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

3.

Example:

is non - linear because

ydx

dysin

!3

sin3y

yy is non – linear

4.

It is Ordinary/partial Differential equation of order… and of degree…, it is

linear / non linear, with independent variable…, and dependent variable….

1st – order differential equation

2. Differential form:

01 ydxdyx.

.0),,( dx

dyyxf),( yxf

dx

dy

3. General form:

or

1. Derivative form:

xgyxadx

dyxa 01

9 Differential Equation

First Order Ordinary Differential

equation

10 Differential Equation

Second order Ordinary Differential

Equation

11 Differential Equation

nth – order linear differential

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

xgyadx

dya

dx

yda

dx

yda

dx

yda

n

n

nn

n

n

012

2

21

1

1 ....

2. nth – order linear differential equation with variable coefficients

xgyxadx

dyxa

dx

ydxa

dx

ydxa

dx

dyxa

n

n

nn

012

2

2

1

1 ......

12 Differential Equation

Differential Equation 13

Solution of Differential Equation

y=3x+c1 , is solution of the 1st order differential equation , c1 is arbitrary constant.

As is solution of the differential equation for every value of c1, hence it is known as general solution.

3dx

dy

Examples

Examples

sin cosy x y x C

2 3

1 1 26 e 3 e ex x xy x y x C y x C x C

Observe that the set of solutions to the above 1st order equation has 1 parameter,

while the solutions to the above 2nd order equation depend on two parameters.

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Families of Solutions 9 ' 4 0yy x

1 19 ' 4 9 ( ) '( ) 4yy x dx C y x y x dx xdx C

2 2

1This yields where .4 9 18

Cy xC C

Example

Solution

The solution is a family of ellipses.

2

2 2 2 2

1 1 1

99 2 2 9 4 2

2

yydy x C x C y x C

Observe that given any point (x0,y0),

there is a unique solution curve of the

above equation which curve goes

through the given point.

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Origin of Differential Equations

Solution

1.Geometric Origin

21 cxcy

02

2

dx

yd

1. For the family of straight lines

the differential equation is

. 2. For the family of curves

2

2x

cey the differential equation is xy

dx

dy

A.

B. xx ececy 3

2

2

1

the differential equation is 06

2

2

ydx

dy

dx

yd

16

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Physical Origin

1. Free falling stone g

dt

sd

2

2

where s is distance or height and

g is acceleration due to gravity.

2. Spring vertical displacement ky

dt

ydm

2

2

where y is displacement,

m is mass and

k is spring constant

3. RLC – circuit, Kirchoff ’s Second Law

Eqcdt

dqR

dt

qdL

12

2

q is charge on

capacitor,

L is inductance,

c is capacitance.

R is resistance and

E is voltage 17

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Physical Origin

1.Newton’s Low of Cooling sTTdt

dT

where dt

dT is rate of cooling of the liquid,

sTT is temperature difference between the liquid ‘T’

and its surrounding Ts

dy

ydt

2. Growth and Decay

y is the quantity present at any time

18 Differential Equation

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Thanking you

Differential Equation 19