Page 43 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes So the two solutions of Legendre’s equations form the general solution
y = A y 1 B y 2 ...(Q)
In particular, if we take constant C to be
0
1.3.5...(2n 1)
C =
0 ! n
in equation (N), we get the solution

1.3.5 (2n 1) n ( n n 1) n 2 ( n n 1)(n 2)(n 3) n 4
x
P n ( ) = n x 2(2n 1) x 2.4 (2n 1)(2n 3) x ... ...(R)
denoted by P n ( ) , and is called Legendre’s function of first kind.
x
Legendre’s Functions of the Second Kind
When n is a positive integer and putting the value of C , as
o
! n
C = ...(S)
o 1.3.5....(2n 1)
in the second solution (P) we get the Legendre’s function of the second kind denoted by Q n ( )
x
i.e.
! n n 1 (n 1) (n 2) n 3 (n 1)(n 2)(n 3)(n 4) n 5
Q n ( ) = x x x ... ...(T)
x
1.3.5...(2n 1) 2.(2n 3) 2.4. (2n 3) (2n 5)
x
As is seen from equation (T), Q n ( ) is an infinite or non-terminating series.
Thus the general solution of Legendre’s equation is
y = A P n ( ) B Q n ( ) ...(U)
x
x
2.2 Rodrigue’s Formula for Legendre Polynomials

An other formula for P n ( ) can be obtained from the Legendre’s differential equation. Here we
x
start with
2
u = (x 1) n ...(A)
du
Then = 2 nx (x 2 1) n 1
dx
2
Multiplying both sides by (x 1) and transposing to left hand side, we get
(x 2 1) du 2nx (x 2 1) n = 0
dx
du
2
or (x 1) 2nx u = 0
dx
Differentiating the above equation with respect to x, we get
2
2 d u du du
(1 x ) 2x 2nu 2nx = 0
dx 2 dx dx
2
2 d u du
(1 x ) 2 2(n 1)x 2nu = 0 ...(B)
dx dx

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