Differential equation

1. Linear Differential Equation
with constant coefficient

2. Ordinary Differential Equations
)
cos(
2
5
:
2
2
x
y
dx
dy
dx
y
d
e
y
dx
dy
Examples
x





Ordinary Differential Equations (ODE) involve one or
more ordinary derivatives of unknown functions with
respect to one independent variable
y(x): unknown function
x: independent variable
2

3. Order of a differential equation
1
2
)
cos(
2
5
:
4
3
2
2
2
2
















y
dx
dy
dx
y
d
x
y
dx
dy
dx
y
d
e
y
dx
dy
Examples
x
The order of an ordinary differential equations is the order
of the highest order derivative
Second order ODE
First order ODE
Second order ODE
3

4. Linear ODE
1
)
cos(
2
5
:
3
2
2
2
2
2
















y
dx
dy
dx
y
d
x
y
x
dx
dy
dx
y
d
e
y
dx
dy
Examples
x
An ODE is linear if the unknown function and its derivatives
appear to power one. No product of the unknown function
and/or its derivatives
Linear ODE
Linear ODE
Non-linear ODE
4
)
(
)
(
)
(
)
(
'
)
(
)
(
)
(
)
(
)
( 0
1
1
1 x
g
x
y
x
a
x
y
x
a
x
y
x
a
x
y
x
a n
n
n
n 



 
 

5. with constant coefficient
The nth
order linear differential equation
The Differential Equation of the form
dn
y dy
a0
dxn
dn1
y dn2
y
a1
dxn1
dxn2
a2 .......an1
dx
an y  Q
Exa mple:
d 3
y  3 d 2
y  6 dy  2 y  sin 5 x
dx3
dx2
dx

6. F ( D ) y  Q
dx
If d  D
o n
n1
Where F (D)  a Dn a Dn1
 a Dn2
1 2  .......  a D  a
dx
dx3 dx2
Example : d3y 3d2y 6dy2ysin5x
(D3y3D2y6Dy2y)Sin5x
(D33D26D2)ySin5x
 F(D)y  Sin5x
F(D)(D33D26D2)

7. Auxiliary Equation(A.E.)
Suppose L.D.E. is F(D)y  Q
A.E . is F ( m )  0
1 2
o
OR a mn
 a mn1
 a mn2
....... an1m an  0
dx
dx3 dx2
Example: d3 y  3d2 y  6 dy  2y  sin5x
 F(D)y  Sin5x
(D33D2 6D2)ySin5x
F(D)(D33D2 6D2)
Hence A.E.is F(m)0m33m2 6m20

8. Complementary Function (C.F.) of L.D.E.
is known as
is known as
General Solution of L.D.E.
The general solution of L.D.E is given by
y = C.F. + P.I
A function of ‘x’which satisfies the L.D.E F(D)y0
complementary function of L.D.E . .
Particular Integral (P.I.) of L.D.E.
A function of ‘x’which satisfies the L.D.E. F(D)yQ
particular integral of L.D.E .
F(D)yQ

9. General Solution of L.D.E.
Where C.F
P.I
Complementary Function
Particular Integral
Suppose L.D.E. is F (D ) y  Q
Complete solution :
y = C.F+P.I

10. Complementary Function
A function of ‘x’ which satisfies the L.D.E
F(D)y = 0
is known as complementary function of
L.D.E . .

11. Determination of C.F.
●
●
Consider the L.D.E . F(D)y = 0
Write A.E. of L.D.E. F(m) = 0
●
● Suppose
are the ‘n’ roots of the auxiliary equation.
n
n1
 a mn
amn1
a mn2
.......a
o 1 2
Solve A.E.
ma 0
m1 , m2 , m3 ,........., mn

13. Case I: (Roots are real)
3
1 2
1 2
n
m x m x m x m x
3 n
If m1,m2,m3,.........,mn are real and distinct
then C.F  c e c e c e .......c e