Introduction to Differential Equations

1.1 Definitions and Terminology

Definition:

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

32 xdx

dy

032

2

aydx

dy

dx

yd

364

3

3

ydx

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

364

3

3

ydx

dy

dx

yd

1

2

3

Degree of Differential Equation

Differential Equation Degree

032

2

aydx

dy

dx

yd

364

3

3

ydx

dy

dx

yd

0353

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:

364

3

3

ydx

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

10Differential Equation Chapter 1

First Order Ordinary Differential equation

11Differential Equation Chapter 1

Second order Ordinary Differential Equation

12Differential Equation Chapter 1

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 ......

13Differential 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),,',,( )( nyyyxF

0))(,),('),(,( )( xxxxF n

Differential Equation Chapter 1 15

Verification of a solution

Verify that the indicated function is a solution of the given DE on the interval (-∞, ∞).

a)

b)

,yxdx

dy 4

16

1xy

0'2'' yyy xxey

Differential Equation Chapter 1 16

Verification of an Implicit Solution

The relation is an implicit solution of DE

2522 yx

y

x

dx

dy

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

3dx

dy

Examples

Examples

sin cosy x y x C

2 31 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.

Families of Solutions 9 ' 4 0yy x

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

2 21This yields where .

4 9 18

Cy xC C

Example

Solution

The solution is a family of ellipses.

2

2 2 2 21 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.

Diff

ere

ntia

l E

qua

tion

C

ha

pte

r 1

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 Problems1.2 Initial-Value Problems

21

Example

22

Interval I of Definition of a Solution

23

2

1y

x c

is a solution of the first-order differential equation 02 2' xyy

If we impose the initial condition y(0)=-1 we will get c = -1 and

therefore the solution will be 2

1.1

yx

We now emphasize the following three distinctions:

• Considered as a function, the domain of is R\{-1,1}.2

1

1y

x

24

• Considered as a solution of the differential equation the interval I of definition of could be taken to be any interval over

which y(x) is defined and differentiable. The largest intervals on which

is a solution are (-∞,-1), (-1,1) and (1,∞).

• Considered as a solution of the initial -value problem

y(0)=-1, the interval I of definition of could be taken to be any

interval over which y(x) is defined, differentiable, and contains the initial

point x = 0; the largest interval for which this is true is (-1,1).

02 2' xyy

2

1

1y

x

2

1

1y

x

,02 2' xyy

2

1

1y

x

25

Let R be a rectangle region in the xy-plane defined by

dycbxa ,

EXAMPLE

26

Solution

27

28