Page 215 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 13.4 Orthogonality of Solutions of Some Equations

(a) Orthogonality of Bessel’s Functions
We know that J (x ) is the solution of Bessel’s equation
n
2
2 d J n ( ) d J n ( ) 2 2
x
x
x x (x n ) J n ( ) = 0
x
dx 2 dx
where n is a positive integer. Putting x , x we have
d J n = 1 d J n
dx dx
2
2
d J n 1 d J n
and 2 = 2 2 ,
dx dx
where is a constant,
2
2 d J n ( x ) d J n ( x ) 2 2 2
x 2 x ( x n ) ( x ) = 0 ...(i)
J
n
dx dx
which may be rewritten as
d d J n (x ) 2 n 2
x x J n ( x ) = 0
dx dx x
which is Sturm Liouville equation for each fixed n i.e.
d d
p ( ) J n ( x ) [ ( ) 1 r ( )]y = 0
q
x
x
x
dx dx
n 2
with p(x) = x , ( ) and ( )r x x and 1 2 .
q
x
x
Since p(x) = 0 for x = 0, it follows that the solution of (i) on an interval 0 x a satisfying the
boundary conditions
J n ( a ) = 0 ...(ii)
form an orthogonal set with respect to the weight p(x) = x.
Let < < ... denote the positive zeros of J (x ), therefore (ii) holds for
1n 2n 3n n 1

a = or = = mn (m = 1, 2, ... n fixed)
mn mn a
d
x
and since J n ( ) is continuous also at x = 0, therefore for each fixed n = 0, 1, 2, .... , the Bessel’s
dx
mn
function J ( ) x (m = 1, 2, ...) with , form a orthogonal set on an interval 0 x a
n mn mn a
with respect to weight function p(x) = x,

a
x J n mn x J n pn x = 0 if p m
0

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