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Introduction to Ordinary

Differential Equations

Chapter 1

Overview

Overview

II. Classification of Solutions

Chapter 1: Introduction to Differential Equations

I. Definitions

Basic Example

Basic Example

Consider

Consider

x

f x e

2

( ) 

x

f x e

' 2

( )  2

( ) 2 ( ) 2 2 0

' 2 2

   

x x

f x f x e e

satisfies the Differential Equation:

satisfies the Differential Equation:

f

2 0

'

y  y 

What is a Differential Equation

What is a Differential Equation

A

A

differential equation (DE)

differential equation (DE) is an equation containing the

is an equation containing the

derivatives of one or more dependent variables with

derivatives of one or more dependent variables with

respect to one or more independent variables.

respect to one or more independent variables.

Classification

Classification

Differential equations (DE)

Differential equations (DE) can be classified by:

can be classified by:

TYPE

TYPE

ORDER

ORDER

LINEARITY.

LINEARITY.

Classification by Type

Classification by Type

Two types of

Two types of Differential equations (DE)

Differential equations (DE) exist

exist :

ORDINARY DIFFERENTIAL EQUATION (ODE).

ORDINARY DIFFERENTIAL EQUATION (ODE).

An equation containing only ordinary derivatives of one

An equation containing only ordinary derivatives of one

or more dependent variables with respect to a

or more dependent variables with respect to a SINGLE

SINGLE

independent variable is said to be an

independent variable is said to be an Ordinary

Ordinary

Differential Equation

Differential Equation (ODE).

(ODE).

PARTIAL DIFFERENTIAL EQUATIONS (PDE).

PARTIAL DIFFERENTIAL EQUATIONS (PDE).

An equation containing partial derivatives of one or more

An equation containing partial derivatives of one or more

dependent variables with respect to

dependent variables with respect to TWO

TWO

or more

or more

independent variables is said to be a

independent variables is said to be a Partial Differential

Partial Differential

Equation

Equation (PDE).

(PDE).

Examples of PDE

Examples of PDE

2

2

2

2

y

u

x

u

t

u

t

u

x

u

2

2

2

2

x

v

y

u





 





3 )

Examples of Orders

Examples of Orders

x

y e

dx

dy

 

3

5

6 0

2

2

  y 

dx

dy

dx

d y

x

y e

dx

dy

dx

d y

3

2

2

is of order 1 (or first-order)

is of order 2

is of order 2

First-order ODEs are occasionally written in differential

First-order ODEs are occasionally written in differential

form :

form :

Remark

Remark

M ( x , y ) dx  N ( x , y ) dy  0

Examples for linear ODEs

Examples for linear ODEs

1 )  y  x  dx  4 x dy  0

2 ) 2   0







y y y

x

y e

dx

dy

x

dx

d y

3

3

xy  y  x



 4

Examples for non-linear ODEs

Examples for non-linear ODEs

 

x

- y y  y  e

 1 ) 1 2

2 ) sin 0

2

2

 y 

dx

d y

2

4

4

 y 

dx

d y

Solution:

Solution:

ODE Order Linearity

dy   xy  cos x  dx  0

6 0

2

2

  

dt

dQ

dt

d Q

2   0

2

 

 

 

 y xy y y xy

e  xy y

y

Linear

Linear

Non-linear

Non-linear

1

2

3

2

Solution:

Solution:

ODE Order Linearity

   sin 



 1  0

2

y  dx  xdy 

2

2

2

1













 

dx

dy

dx

d y

Linear

Non-linear

Non-linear

1

1

2

  

2

  sin

