Page 59 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes where
1
2r 1
C = f ( ) ( )dx
x
x
P
r r
2
1
1
(2r 1)
P
C = 1. ( )dx for r 1, 2,... ...(iii)
x
r
r 2
0
Putting r 0, 1, 2, 3, ....
1 1 1
1 1 1 1
x
C = P ( )dx 1.dx x
0 2 0 2 2 2
0 0 0
1 1 2 1
3 3 3 x 3
C = P ( )dx xdx .
x
1 1
2 2 2 2 4
0 0 0
1
5
C = P 2 ( ) dx
x
2 2
0
1
5 1
= (3x 2 1)dx
2 2
0
1
5 3x 3
= x 0
4 3
0
1
7
C = P ( )dx
x
3 2 3
0
1
7 1
x
= (5x 3 3 )dx
2 2
0
7 5x 4 3x 2 1
=
4 4 2
0
7 5 3
=
4 4 2
7 5 6 7
=
4 4 6
1 3 7
x
So f ( ) = P 0 ( ) P 1 ( ) P 3 ( ) ...
x
x
x
2 4 16

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