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



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







































(c) John H. Mathews 2004