Example 8.  In Example 7, the boundary condition for the D.E. is  [Graphics:Images/FrobeniusSeriesMod_gr_490.gif],  i.e. the drum head has radius  [Graphics:Images/FrobeniusSeriesMod_gr_491.gif].
Thus the parameter  [Graphics:Images/FrobeniusSeriesMod_gr_492.gif]  must be chosen to be a root of the Bessel function.
The zeros do not have a simple formula. However it is known that they are "close to" multiples of  [Graphics:Images/FrobeniusSeriesMod_gr_493.gif].  
Verify this and find the first five zeros.
Solution 8.

Multiples of  [Graphics:../Images/FrobeniusSeriesMod_gr_494.gif]  will be sufficiently close to be starting values for Mathematica's  FindRoot  subroutine.
The first root is

[Graphics:../Images/FrobeniusSeriesMod_gr_495.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_496.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_497.gif]

We can put all five of them in an array called "roots."  Then redraw the graph with horizontal axis ticks at the integers.

[Graphics:../Images/FrobeniusSeriesMod_gr_498.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_499.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_500.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004