Mathematical Methods for Physicists: A concise introduction - Site Map

BESSEL'S EQUATION
Figure 7.5.
A ¯exible chain.
from which we obtain
@2 y
@t 2 ˆ @
@x
T @y
:
@x
Now T ˆ gx. Substituting this into the above equation for y, we obtain
@ 2 y
@t 2 ˆ g @y
@x ‡ gx @2 y
@x 2 ;
where y is a function of two variables x and t. The ®rst step in the solution is to
separate the variables. Let us attempt a solution of the form y…x; t† ˆu…x† f …t†.
Substitution of this into the partial di€erential equation yields two equations:
f 00 …t†‡! 2 f …t† ˆ0; xu 00 …x†‡u 0 …x†‡…! 2 =g†u…x† ˆ0;
where ! 2 is the separation constant. The di€erential equation for f …t† is ready for
integration and the result is f …t† ˆcos…!t †, with a phase constant. The
di€erential equation for u…x† is not in a recognizable form yet. To solve it, ®rst
change variables by putting
x ˆ gz 2 =4;
w…z† ˆu…x†;
then the di€erential equation for u…x† becomes Bessel's equation of order zero:
Its general solution is
or
zw 00 …z†‡w 0 …z†‡! 2 zw…z† ˆ0:
w…z† ˆAJ 0 …!z†‡BY 0 …!z†
r
r
x
x
u…x† ˆAJ 0 2! ‡ BY
g 0 2! :
g
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