Page 132 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 7: General Properties of Solutions of Linear Differential Equations of Order n

7.4 Particular Integral Notes

1
Let Q ...(i)
f ( )
D
denote som e function of x which when operated upon by f(D) gives Q. This function of x is a
particular solution of the differential equation.
f ( )y = Q ...(ii)
D
1
As f(D) and f(D) are inverse operations, therefore
Q
D {D 1 ( )} = Q (Particular case)
d 1
Q
or {D ( )} = Q
dx
1
D (Q) = Q dx

1
Example: Properties of .
f ( )
D
1. If Q u 1 u 2 u 3  u n then

1 1 1 1
Q u 1 u 2  u n .
D
D
f ( ) f ( ) f ( ) f ( )
D
D
1 1
2. (k Q ) . k Q . where k is a constant
D
f ( ) f ( )
D
1
3. can be resolved into factors.
f ( )
D
1
4. can be broken into partial fractions.
D
f ( )
1
5. Q is a particular integration.
D
f ( )
1 x ex
To show that Q e e Qdx
D
1
Let Q = V
(D )
Therefore (D ) V = Q
d
or V = Q
dx
This is a linear differential equation. The solution is

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