Chapter 1

Introduction to Differential Equations

A differential equation (DE) is an equation containing the

derivatives of one or more dependent variables with respect

to one or more independent variables.

Definition:

4 3dy

xdx

= −2

26 0

d yy

dx+ =

2 2

2 24 0

y y

t x

∂ ∂− =

∂ ∂

Ordinary Differential Equations (ODEs):

A differential equation is an ODE if it contains only ordinary

derivatives of one or more dependent variables with respect to

a single independent variable.

Types of Differential Equations

25 −= xdx

dy0sin

2

2

=+ ydx

yd02

2

2

2

=

+

dx

dy

dx

yde y

Partial Differential Equations (PDEs):

A differential equation is a PDE if it contains the partial

derivatives of one or more dependent variables with respect to

two or more independent variables.

Types of Differential Equations

02

2

2

2

=∂

∂+

∂

∂

y

u

x

u0

2

2

2

2

2

2

=∂

∂+

∂

∂+

∂

∂

z

u

y

u

x

u

The order of a differential equation (either ODE or PDE) is the

order of the highest derivative that appears in the equation.

Order of Differential Equations

xdx

dycos=

xdx

dy

dx

yd76

4

2

2

=

+

( ) 422 22 yxyeyx x +=′′+′′′

First Order

Second Order

Third Order

Linearity of Ordinary Differential Equations

An nth-order ODE is linear if it can be written in the form:

The dependent

variable and all its

derivatives are of the

first degree (the

power of each term

involving y is one).

The coefficients are

merely constants or

they depend only

on the independent

variable x .

The dependent

variable y and all its

derivatives are linear

functions.

)()()()(...)()( 012

2

21

1

1 xgyxadx

dyxa

dx

ydxa

dx

ydxa

dx

ydxa

n

n

nn

n

n =+++++−

−

−

Example 1: Which of these differential equations are

linear?

xyyx ln53 =−′

( ) xey

yx =′

+′′2

sin

xxydx

dy

dx

yd=+

+ 4

3

2

2

253 xyyy =+′

0sin2

2

=+ ydx

yd

Answer: Which of these differential equations are linear?

xyyx ln53 =−′

( ) xey

yx =′

+′′2

sin

xxydx

dy

dx

yd=+

+ 4

3

2

2

Both y and y’ are linear. The differential equation is linear

Both y and y’ are linear. The differential equation is linear

The term (dy/dx) is not linear. The differential equation is

non-linear

Answer: Which of these differential equations are linear?

The coefficient of y’ does not depends on x. The

differential equation is non-linear

The term sin y is not a linear function. The differential

equation is non-linear

253 xyyy =+′

0sin2

2

=+ ydx

yd

State the order and linearity of each differential equation below:

Exercise 1:

Exercise 2:

Any function ϕ, defined on an interval I and possessing at least n

derivatives that are continuous on I , which when substituted into

an nth-order ODE reduces the equation to an identity, is said to be

a solution of the equation on the interval.

A function ϕ(x) is a solution if it is differentiable and satisfies the

DE for all x in I . The interval I is the domain of the solution.

Solution of an ODE

Verify that a function y(x) = c1e4x + c2e

3x , (where c1 and c2 are

arbitrary constants), is a solution of the differential equation

y’ – 4y = -e3x on the interval (-∞, ∞).

Example 1:

Verify that a function y(x) = c1sin2x + c2cos2x , (where c1 and c2

are arbitrary constants), is a solution of the differential equation

y” + 4y = 0 on the interval (-∞, ∞).

Example 2:

Exercise 3:

An explicit solution is a solution in which the dependent

variable is expressed solely in terms of the independent

variable and constants i.e: y = f(x) + c

Types of Solutions

An implicit solution is a solution which is implicitly given in

the form of g(x,y) = c.

A solution of a first-order DE F(x, y, y’) = 0 usually

contains a single arbitrary constant or parameter c. It

represents a set G(x,y,c) = 0 of solutions called a one-

parameter family of solutions.

An nth order DE F(x, y, y’,….,y(n)) = 0 possesses an

n-parameter family of solutions G(x,y,c1,c2,…,cn) = 0

Types of Solutions

A particular solution is a solution of a DE that is free of

arbitrary parameters.

Example: y = x2 + 3 is a particular solution of the DE

y’= 2x.

Types of Solutions

A singular solution is an extra solution of a DE that cannot be

obtained by specializing any of the parameters in the family

of solutions.

A differential equation together with an initial condition is

called an initial-value problem (or IVP).

Initial Value Problems

First-Order IVP:

0 0( , ) , ( )dy

f x y y x ydx

= =

Second-Order IVP:

2

0 0 0 12( , , ) ; ( ) , ( )

d yf x y y y x y y x y

dx′ ′= = =

Find a solution of the first-order IVP:

if the general solution of the differential equation is

Example 3: Solution of Initial Value Problems

2

1

( )y

x c=

+

2 12 =0 ; (2)

3y xy y′ + =

(Hint: use the intial condition to find c)

Find a solution of the second-order IVP:

if the general solution of the differential equation is

Example 4:

1 2

x xy c e c e−= +

=0 ; (1) 0 , (1)y y y y e′′ ′− = =

(Hint: use the intial conditions to find c1 and c2 )