Page 38 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 2: Legendre’s Polynomials

Notes
2
d y d dy 2 d 2 dy
= x r
dx 2 dx dx dr dr
2
dy 4 d y
2
= r 2r r 2
dx dr
The equation (F) becomes
2
4 d y 2r 2 dy ( n n 1)
r y = 0
dr 2 1 dx 1
r 1 2 1 2
r r
2
4 d y 2r 3 dy ( n n 1)r 2
r y = 0
dr 2 (x 1) dx (r 2 1)
2
d y 2 dy ( n n 1)
or 2 2 2 2 y = 0 (I)
dr ( r r 1) dr r (r 1)
Thus r = 0 or x = is a regular singular point of the differential equation (Legendre’s). Thus we can
1
find a solution of Legendre’s equation in terms of a power series in x as well as in powers of .
x

2.1.1 Power Series Solution of Legendre’s Equation in Ascending Powers
of x

2
2 d F dF
(1 x ) 2 n 2x n ( n n 1)F n = 0 ...(A)
dx dx
As in the case of Bessel’s differential equation we assume a solution of the form:

F n = x s C x r
r
r 0

or F n = C x r s ...(B)
r
r 0
For (B) to be a solution of (A) it is necessary that when equation (B) is substituted into (A), the
coefficients of every power of x vanish. So we have

(1 x 2 ) (r s )(r s 1)C x r s 2 2x C r (r s )x r s 1 ( n n 1) C x r s 0
r
r
r 0 r 0 r 0
or (r s )(r s 1)C r (x r s 2 x r s ) 2C r (r s )x r s x r s ( n n 1)C r 0
r 0

or (r s )(r s 1)C x r s 2 C x r s [ (n 1) 2(r s ) (r s )(r s 1)] 0
n
r
r
r 0
(r s )(r s 1)C x r s 2 C x r s (n r s )(n r s 1) ...(C)
r
r
r 0

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