Page 136 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

Notes
1
,
To evaluate sinax where ( )f D P D n P D n 1 ...P .
D
f ( ) 0 1 n
Case I. When f(D) contains even powers of D
Let ( f D 2 ) = P (D 2 n P (D 2 n 1  P
)
)
0 1 . n
2
2
We notice that D sin ax = a sin ax.
2 2
4
D sin ax = ( a ) sin ax
2 3
6
D sin ax = ( a ) sin ax
.................................................
.................................................
2 n
2 n
(D ) sin ax = ( a ) sin ax
Therefore ( f D 2 )sin ax = P (D 2n P D 2n 2  P )sin ax
0 1 n
or ( f D 2 )sin ax
= P D 2n sinax P D 2n 2 sin ax  P sin ax
0 1 n
= P ( a 2 n ( a 2 n 1 sin ax ... P sin ax
) sin ax P
)
0 1 n
= ( f a 2 )sin ax .
1
Operating on both sides with 2 , we have
( f D )

1 2 1 ( f a 2 )sin ax
( f D )sin ax = 2
D
( f D 2 ) f ( )
1
2
or sin ax = ( f a ). 2 sin ax .
( f D )
Dividing both sides by (f a 2 ), we have

1 1 sin ax .
( f D 2 ) sin ax = ( f a 2 )
Case II. When f(D) contains odd powers of D.

Let it be put in the form f (D 2 ) Df (D 2 ); then
1 2
1 1
sin ax = 2 sin ax
D
f ( ) f 1 (D ) D f 2 (D )
1
= 2 2 sin ax
( f a ) D f 2 ( a )

1
= sin ax say
m nD

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