Page 217 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Putting = x,
d
x J n ( x ) n J n ( x ) = x J ( x )
( d x ) n 1

or x J 1 ( x ) n J ( x ) = x J ( x )
n n n 1
Substituting in (v), we have

a
2 a a
2
n J ( x ) x J ( x ) = 2 2 n J 2 ( x ) 2 2 x J 2 ( x )dx
x
n n 1 n n
0 0
0
J
If mn , then n ( a ) J n mn a 0, and
J (0) 0, for n = 1, 2, ...,
Since n
then we have
a
2 2 2 2 2 x J 2 x dx
mn a J n 1 mn a = mn n mx
0

= 2 2 mn J n ( mn ) x 2 {since weight = x}
Thus

2 a 2 2
J n ( mn ) x = J n 1 ( mn ) a
2
a 2 2
= J n 1 ( mn )
2
where = a
mn mn
a
2
f
x
So C = 2 2 x J n ( mn x ) ( )dx ...(vi)
n a J ( )
1
mn
n
0
and = mn , for m = 1, 2, 3....
mn a
Thus generalized Fourier Bessel series is given by (iii) with the coefficient C given by (vi).
n
(b) Orthogonality of Legendre Polynomials
The Legendre’s differential equation

(1 x 2 )y 2xy ( n n 1)y = 0

may be written as

[(1 x 2 ) ] y = 0 ...(i)
y
where = n (n + 1),

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