Page 37 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes F
sin F n = sin 2 n
x

F
sin F n = sin 2 n
x
2
F n 3 d F n
= 2 sin cos sin 2
dx dx
Substituting in equation (D) we have
2
F 2 d F
( n n 1)F n 2cos n sin 2 = 0
x dx
2
dF x 2 d F
or ( n n 1)F n 2x (1 x ) 2 = 0
dx dx
Rewriting it as:
2
2 d F n dF n
(1 x ) 2 2x ( n n 1)F n = 0 ...(E)
dx dx
This equation (E) is known as Legendre’s differential equations. The solution of equation (E) for
positive integer values of n are known as Legendre Polynomial.
Putting Legendre equation in Fuchs form we have

2
d F n 2x dF n ( n n 1) F
dx 2 (1 x 2 ) dx (1 x 2 ) n = 0 ...(F)

dF n
Here let coefficients of and F be
dx n
2x
x
p ( ) = ..(G)
(1 x 2 )
( n n 1)
x
q ( ) = 2
(1 x )
At x 1 and x 1, both ( )p x and ( )q x have poles of the first order. So the points x = 1 and
x = 1 are regular singular points of the Legendre’s equations. Let us investigate the behaviour
of the equation for x . For this purpose let us put
1
x = , F n y ...(H)
r

dy dy dr 2 dy
= r
dx dx dx dr

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