Page 135 - Advanced engineering mathematics

3.7 Polynomial Coefficients 115

We will use the Laplace transform to derive solutions of Bessel’s equation. Consider first the
case n = 0. Bessel’s equation of order zero is

ty + y + ty = 0.

Apply L to obtain

L[ty ]+ L[y ]+ L[ty]= 0.

Then
d
d 2

− s Y(s) − sy(0) − y (0) + sY(s) − y(0) + (sY(s) − y(0)) = 0.
ds ds
This is
2

−2sY − s Y + sY − Y = 0

or
2
−sY − (1 + s )Y = 0.

This is a separable differential equation for Y. Write
Y s
=− .
Y 1 + s 2
Integrate to obtain

1 2 2 −1/2
ln|Y|=− ln(1 + s ) + c = ln((1 + s ) ) + c.
2
Take the exponential of both sides of this equation to write
C
c
2 −1/2
Y(s) = e (1 + s ) = √
1 + s 2
in which C = e is constant. We have to invert this. First rewrite
c
−1/2
C 1

Y(s) = 1 + .
s s 2
The reason for doing this is to invoke the binomial series, which in general has the form
k(k − 1)
k 2
(1 + x) = 1 + kx + x
2!
k(k − 1)(k − 2) 2
+ x + ···
3!
k
∞

m
= x for |x| < 1.
m
m=0
Here

k 1 for m = 0,
=
m k(k−1)···(k−m+1) for m = 1,2,···.
m!
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:14 THM/NEIL Page-115 27410_03_ch03_p77-120