Ordinary Differential Equations
Dr. Ahmed Elmoasry
ordinary differential equations
Definition:
A differential equation is an equation containing an unknown function
and its derivatives.
Examples:.
1.
dy
 2x  3
dx
2.
d2y
dy

3
 ay  0
2
dx
dx
3.
d 3 y  dy 
    6y  3
3
dx
 dx 
4
y is dependent variable and x is independent variable,
and these are ordinary differential equations
Partial Differential Equation
Examples:
1.
 2u  2u
 2 0
2
x
y
u is dependent variable and x and y are independent variables,
and is partial differential equation.
2.
 4u  4u
 4 0
4
x
t
3.
 2 u  2 u u
 2 
2
t
x
t
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
dy
 2x  3
dx
d2y
dy
 3  9y  0
2
dx
dx
4
d 3 y  dy 
    6y  3
3
dx
 dx 
ORDER
1
2
3
Degree of Differential Equation
The degree of a differential equation is power of the highest
order derivative term in the differential equation.
Differential Equation
d2y
dy
 3  ay  0
2
dx
dx
Degree
1
4
d 3 y  dy 
    6y  3
3
dx
 dx 
3
 d 2 y   dy 
 2      3  0
 dx   dx 
1
5
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:
d2y
dy

3
 9 y  0.
1.
2
dx
dx
is linear.
Example: 2.
4
d 3 y  dy 
    6y  3
3
dx
 dx 
is non - linear because in 2nd term is not of degree one.
Example: 3.
d2y
dy
3
x

y

x
dx
dx 2
2
is non - linear because in 2nd term coefficient depends on y.
Example: 4.
dy
 sin y
dx
is non - linear because
y3
sin y  y 

3!
is non – linear
Examples of linear diff Eq.
x dy  ydx  0
y  2 y  y  0
3
2
d
y
d
y
dy
3
2
x
x

x

3
x

5
y

e
3
2
dx
dx
dx
Examples of non linear diff Eq.
Coefficient
depends on y
yy  2 y  x
Power not
1
3
d y
2

y

0
3
dx
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
1. Derivative form:
dy
a1  x 
 a0 x  y  g x 
dx
2. Differential form:
1  xdy  ydx  0
.
3. General form:
dy
 f ( x, y )
dx
or
Differential Equation Chapter 1
dy
f ( x , y , )  0.
dx
11
First Order Ordinary Differential
equation
Differential Equation Chapter 1
12
Second order Ordinary Differential
Equation
Differential Equation Chapter 1
13
nth – order linear differential
equation
1. nth – order linear differential equation with constant coefficients.
dny
d n 1 y
d2y
dy
a n n  a n 1 n 1  ....  a 2 2  a1
 a0 y  g x 
dx
dx
dx
dx
2. nth – order linear differential equation with variable coefficients
dy
d n1 y
d2y
dy




a n x   a n 1 x 

......

a
x

a
x
 a0 x  y  g x 
2
1
n
2
dx
dx
dx
dx
Differential Equation Chapter 1
14
Solution of Differential Equation
Differential Equation Chapter 1
15
Examples
y=3x+c
, is solution of the 1st order
differential equation dy/dx=3, c1 is arbitrary constant.
As is solution of the differential equation for every
value of c1, hence it is known as general solution.
Examples
y   sin  x   y   cos  x   C
y   6 x  e x  y   3 x 2  e x  C1  y  x 3  e x  C1x  C2
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.
Examples
yx
4
16
y  xe ,
x
,
dy
1/ 2
y 
 xy
dx
y  2 y  y  0
Differential Equation Chapter 1
17
Implicit & Explicit solutions
dy
x
 ,
dx
y
y  c cos 4 x,
x  y  4  0.
2
2
y  16 y  0.
x
x
y  e , y  e , y  c1e , y  c2e ,
x
x
x
and y  c1e  c2e .
y  y  0.
x
Differential Equation Chapter 1
18
Chapter 1
Differential Equation
Families of Solutions
Example
9 yy ' 4 x  0
Solution
  9yy ' 4 x  dx  C
1
  9 y ( x )y '( x )dx   4 xdx  C1
9y 2
  9ydy  2x  C1 
 2x 2  C1  9y 2  4 x 2  2C1
2
2
C1
y 2 x2
This yields

 C where C  .
4
9
18
Observe that given any point (x0,y0),
there is a unique solution curve of the
above equation which curve goes
through the given point.
The solution is a family of ellipses.