Page 67 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
Now if a 1 0, then we have

y = a x r
r
r 0
= a 0 a x a x 2 a x 3
1
3
2
2 2 2 2 ( 2) 4 ( 2) m ( 2) ( 2 m 2) 2m
= a 0 1 x x x
m
2! 4! (2 )!
2
2( 1) 3 2 ( 1)( 3) 5
a x x x
1
3! 5!
m
( 2) ( 1)( 3) ( 2m 1) 2m 1
x ...(viii)
(2m 1)!
and if a = 0, then we have
1
2 2 2 2 ( 2) 4 ( 2) m ( 2) ( 2m 2) 2m
y = a 0 1 x x x ...(ix)
2! 4! (2 )!
m
= y (say).
1
Case II: When k = 1, from (vi), we have

2(r 1) 2
a = a r .
r 2 (r 3)(r 2)

Putting r = 1, 3, ... etc.
a = a = ... = 0 (each).
3 5
Since in this case from (iv), a = 0
1
Putting r = 0, 2, 4, ... etc.

2 2 2( 1)
a = a 0 a 0
2 3.2 3!
6 2 2( 1)( 3)
a = a 2 a 0
4 5.4 3!
and so on.

m
( 2) ( 1)( 3) ( 2m 1)
a = a 0
2m (2m 1)!
we have

y = a x r 1
r
r 0
= a x a x 3 a x 5 a x 2m 1
0 2 4 2m

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