Introduction to Differential Equations, Lecture notes of Differential Equations

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Introduction to Differential

Equations

1.1 Definitions and

Terminology

Partial Differential Equation

Examples:

2

2

2

2

y

u

x

u

0

4

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

Order of Differential Equation

The order of the differential equation is order of the highest

derivative in the differential equation.

Differential Equation ORDER

 2 x  3

dx

dy

3 9 0

2

2

  y 

dx

dy

dx

d y

4

3

3

 y

dx

dy

dx

d y

1

2

3

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:

6 3

4

3

3

  











 y

dx

dy

dx

d y

is non - linear because in 2

nd

term is not of degree one.

2

2

  y 

dx

dy

dx

d y

Example:

is linear.

Example:

3

2

2

2

x

dx

dy

y

dx

d y

x  

is non - linear because in 2

nd

term coefficient depends on y.

Example:

is non - linear because

y

dx

dy

sin

sin

3

y

y y

is non – linear

1st – order differential equation

2. Differential form:

 1  x  dy  ydx  0

.

dx

dy

f x y

f ( x , y )

dx

dy



3. General form:

or

1. Derivative form:

  a  x  y g  x 

dx

dy

a x  

1 0

Differential Equation Chapter 1

First Order Ordinary Differential equation

11

Differential Equation Chapter 1

nth – order linear differential

equation

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

a y g  x 

dx

dy

a

dx

d y

a

dx

d y

a

dx

d y

a

n

n

n n

n

n

     





 1 0 2

2

2 1

1

1

....

2. nth – order linear differential equation with variable coefficients

        a  x  y g  x 

dx

dy

a x

dx

d y

a x

dx

d y

a x

dx

dy

a x

n

n

n n

     



 1 0 2

2

2

1

1

......

13

Differential Equation Chapter 1

Differential Equation Chapter 1

14

Solution of an Ordinary Differential Equation

Definition:

A solution of an nth-order ODE

is a function that possesses at least n derivatives and for which

for all x in I.

Interval of definition : You cannot think solution of an ODE without

simultaneously thinking interval. The interval I is called interval of

definition , the interval of existence , the interval of validity , or the

domain of solution.

( , , ', , ) 0

( )



n

F x y y  y

( , ( ), '( ), , ( )) 0

( )

F x x x x 

n

   

Differential Equation Chapter 1

16

Verification of an Implicit Solution

The relation is an implicit solution of DE

2 2

x  y 

y

x

dx

dy

 

y=3x+c , is solution of the 1

st

order

differential equation , c is arbitrary constant.

As y is a solution of the differential equation for every

value of c, hence it is known as general solution.

 3

dx

dy

Examples

Examples

   

y  sin x  y  cos x  C

2 3

1 1 2

6 e 3 e e

x x x

y x y x C y x C x C

Observe that the set of solutions to the above 1

st

order equation has 1 parameter,

while the solutions to the above 2

nd

order equation depend on two parameters.

Families of Solutions

A solution containing an arbitrary

constant is called one-parameter family

of solutions.

A solution of a DE that is free of arbitrary

parameters is called a particular

solution.

19

Differential Equation Chapter 1

20

1.2 Initial-Value Problems

1.2 Initial-Value Problems