Page 12 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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Unit 1: Bessel s Functions

2 d 2 dy Notes
= r r
dr dr
2
dy 2 d y
2
= r 2r r 2
dr dr
2
2
d y 4 d y 3 dy
= r 2r
dx 2 dr 2 dr
,
So the Bessel s equation becomes
2
1 4 d y 3 dy 1 2 dy 1 2
r 2r r n y = 0
r 2 dr 2 dr r dr r 2
2
2 d y dy 1 2
Or r 2 r 2 n y = 0
dr dr r
2
d y 1 dy 1 n 2 y = 0
dr 2 r dr r 4 r 2
,
Thus r 0 or x = is an irregular singular point. Since the singular points for the Bessel s
,
equation are only 0 and , therefore we can get a series solution of the Bessel s equation in
powers of x which converges for 0 < x < . According to Fuchs theorem, the point is regular
singular point provided p(x), q(x) satisfy conditions (C).
Fuchs theorem states that for x = x to be a regular singular point, it is necessary and sufficient
0
that p(x) has at most a pole of order 1 and q(x) at most a pole of order 2.
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1.3.1 Series Solution of Bessel s Differential Equation

,
Bessel s differential equation is
2
2 d y dy 2 2
x 2 x (x n )y = 0 ...(10)
dx dx
Here we shall apply Frobenius method which assumes the solution to be of the form

r
y = x k x c r ...(11)
r 0
Substituting in equation 10 we have

C r (k r )(k r 1)x k r C r (k r )x k r C r (x 2 n 2 )x n r 0
r 0

C x k r (k r )(k r 1) (k r ) (x 2 n 2 ) 0
or r ...(12)
r 0
k
Equating to zero the lowest power of x i.e. x to zero we have
C 0 ( k k 1) k n 2 = 0

or C 0 k 2 n 2 = 0 ...(13)

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