Frobenius Series Solution

Background.

    Consider the second order linear differential equation
    
(1)        
.

Rewrite this equation in the form   
,  then use the substitutions  
  and  
  and rewrite the differential equation (1) in the form  

(2)        
.

Definition (Analytic).  The functions
and
are analytic at
if they have Taylor series expansions with radius of convergence
and  
, respectively.  That is  

        
  which converges for  

    and
        
  which converges for  

Definition (Ordinary Point).  If the functions
and
are analytic at
, then the point
is called an ordinary point of the differential equation  

        
.

Otherwise, the point
is called a singular point.  

Definition (Regular Singular Point).  Assume that
is a singular point of (1) and that  
and  
are analytic at
.

They will have Maclaurin series expansions with radius of convergence
and  
, respectively.  That is  

        
  which converges for  

    and
        
  which converges for  


Then the point  
is called a regular singular point of the differential equation (1).

Method of Frobenius.

    This method is attributed to the german mathemematican Ferdinand Georg Frobenius (1849-1917 ).  Assume that
  is regular singular point of the differential equation

    
.  

    
A Frobenius series (generalized Laurent series) of the form  

        
  

can be used to solve the differential equation.  The parameter
must be chosen so that when the series is substituted into the D.E. the coefficient of the smallest power of  
is zero.  This is called the indicial equation.  Next, a recursive equation for the coefficients is obtained by setting the coefficient of  
  equal to zero.  Caveat: There are some instances when only one Frobenius solution can be constructed.

Definition (Indicial Equation).  The parameter
in the Frobenius series is a root of the indicial equation

        
.

Assuming that the singular point is  
, we can calculate
as follows:

        

and
        

 

The Recursive Formulas.  

    For each root
of the indicial equation, recursive formulas are used to calculate the unknown coefficients
.  This is custom work because a numerical value for
is easier use.  

Application of the Vibrating Drum

    The two dimensional wave equation is   
,  

in rectangular coordinates it is   
,  

and in polar coordinates it is   
.

     Consider a drum head that a flexible circular membrane of radius
.  Assume that it is struck in the center and this produces radial vibrations only where the displacement depends only on time
and distance
from the center.  Then  
satisfies the D.E.
      
    
.

Surface equation for the vibrating drum.

    The solution we are seeking in Example 7 is  
where the boundary condition  
requires that  
,  hence  
. Therefore the fundamental solutions to the wave equation for the drum head is  

    
,  for  n = 1,2,3.