Page 134 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

Notes
A e 1 x e 1 x Qdx A e 2 x e 2 x Qdx
2
1
 A e n x e n x Qdx .
n

1 x
To evaluate e , where
D
f ( )
f(D) = P D n P D n 1  P n ,
1
0
and ( )f 0.

We know that D (e ax ) = ae ax

2 ax
D 2 (e ax ) = a e
...............................
...............................

n
D n (e ax ) = a e ax .
Therefore,

f ( )e ax = (P D n P D n 1  P n ) e ax
D
0
1
n ax
e
= P D e P D n 1 ax  P e ax
0
n
1
n
e
= P a e ax P a n 1 ax  P e ax
1
0
n
= (P a n P a n 1  P n )e ax
0
1
a
Now, ( )f D e ax f ( )e ax .
1
Operating upon both sides with we have
f ( )
D
1 ax 1 ax
f ( )e = f ( )e ,
a
D
D
D
f ( ) f ( )
1
e ax = f ( ) e ax
a
f ( )
D
e ax 1
= e ax , provided ( )f a 0.
a
f ( ) f ( )
D
Illustrative Examples
Example 1: Solve the following equation
(D 2 3D 2)y = e 5x .

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