Frobenius Series Solution

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.

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