Page 137 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2 2
[where m f 1 ( a ),n f 2 ( a )]

1 1
= (m nD ) . sinax
(m nD m nD
)
1
Since (m nD ), are inverse operations.
(m nD )

1
= (m nD ) sin ax
2
(m 2 n D 2 )
1
= (m nD ) 2 2 2 sin ax
m n a

m sin ax na cos ax
=
2
m 2 n a 2
a
f ( a 2 )sin ax f ( a 2 ) cosax
= 1 2 2 2
f 1 ( a 2 ) a 2 f 2 ( a 2 )

1
Notes Similar results are true for cos ax.
D
f ( )
Illustrative Examples

2
Example 1: Solve: (D + D + 1) y = sin 2x.
3 3
Solutions: Here C.F. = e x /2 c 1 cos x c 2 sin x
2 2

1
P.I. = 2 sin2x
D D 1
1
= 2 sin2x
(2) D 1

1
= sin2x
D 3
D 3
= 2 sin2x
D 9

x
D (sin2 ) 3sin2x
=
4 9
1
x
= (2cos2x 3sin2 )
13

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