Page 47 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
8 2 3 20 2 108
x
x
= P 4 ( ) 2 P 3 ( ) x x x
35 5 5 7 35
8 4 20 2 1 108
x
x
= P 4 ( ) P 3 ( ) x x
35 5 7 5 35
8 4 20 2 1 1 108
x
x
x
= P 4 ( ) P 3 ( ) P 2 ( ) x
35 5 7 3 3 5 35
8 4 40 1 224
x
x
x
= P 4 ( ) P 3 ( ) P 2 ( ) x
35 5 21 5 105
8 4 40 1 224
x
x
x
x
x
= P 4 ( ) P 3 ( ) P 2 ( ) P 1 ( ) P 0 ( )
35 5 21 5 105
1 1
Example 3: Prove 1 P 1 (cos ) P 2 (cos ) ...
3 3
... log 1 sin sin
2 2
Solution: From the generating function, we have
n
h P ( ) = (1 2hx h 2 1/2 ...(i)
x
n )
n 0
Integrating w.r.t. h from 0 to 1, we get
1 1 dh
n
x
h P n ( ) dh = ...(ii)
0 0 2
n 0 (1 2hx h )
Replacing x by cos on both sides, (ii) gives
1 1 dh
n
P (cos ) h dh =
n 2
h
n 0 0 0 (1 2 cos h )
h n 1 1 1 dh
or P (cos ) =
n
n 0 n 1 0 0 (h cos ) 2 sin 2
P n (cos )
or n 1 = log(h cos ) (h cos ) 2 sin 2
n 0 0
= log{(1 cos ) [(1 cos ) 2 sin 2 ]} log (1 cos )

= log{(1 cos ) [2(1 cos )}] log(1 cos )

(1 cos ) 2 (1 cos )
= log
(1 cos )

{(1 cos )} {(1 cos )} 2 {(1 cos )}
= log
{(1 cos )} {(1 cos )}

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