Page 367 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes or from (13)

2 d 2 d
(1 x ) 2 (m 1) ( n n 1) m (m 1) = 0 ...(14)
x
dx 2 dx
Let us put

w = (1 x 2 m /2 (1 x 2 m /2 d m m P n ( ) ...(15)
x
)
)
dx
m
then = (1 x 2 ) 2 w

d m m 1 m dw
x
= ( 2 )(1 x 2 ) 2 w (1 x 2 ) 2
dx 2 dx
2
d 2 (1 x 2 ) 2 m 1 w mx ( 2 ) m 1 (1 x 2 ) 2 m 2 w 2mx (1 x 2 ) 2 m 1 dw (1 x 2 ) 2 m d w
x
dx 2 = m 2 dx dx 2
m d w m 1 dw m 2
2
= (1 x 2 ) 2 2 2mx (1 x 2 ) 2 (1 x 2 ) 2 w m (1 x 2 ) mx 2 (m 2)
dx dx
Substituting in equation (14) we have

m 1 d w m dw m 1
2
(1 x 2 ) 2 2mx (1 x 2 ) 2 (1 x 2 ) 2 m mx 2 (m 1) w
dx 2 dx
m 1 m dw
x
2 (m 1)mx (1 x 2 ) 2 w 2 (m 1)(1 x 2 ) 2
x
dx
m
( n n 1) m (m 1) (1 x 2 ) 2 w = 0
m
Dividing by (1 x 2 ) 2 we have
2
2
2 d w dw 2x m (m 1) m mx 2 (m 1)
(1 x ) 2x w ( n n 1) m (m 1) = 0
dx 2 dx (1 x 2 ) (1 x 2 )
2
2 d w dw m 2
(1 x ) 2x ( n n 1) w = 0 ...(16)
dx 2 dx (1 x 2 )
The equation (16) is same as equation (9) where
= w and µ = x
Thus the solution of equation (9) is given by
m m d m
)
= w (1 2 2 (1 2 2 P n ( ) P n m ( ) ...(17)
)
dx m
µ
Where P m ( ) is known as associated Legendre polynomial. Hence the solution of Laplace
n
differential equation is given by (for = m)
V = A r n B r n 1 A e im B e im P m n ( ) ...(18)

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