We begin our study of the method of Frobenius for finding series solutions of linear second order differential equations.

The Method of Frobenius I

In this section we begin to study series solutions of a homogeneous linear second order differential equation with a regular singular point at , so it can be written as

where , , are polynomials and .

We’ll see that (eq:7.5.1) always has at least one solution of the form where and is a suitably chosen number. The method we will use to find solutions of this form and other forms that we’ll encounter in the next two sections is called the method of Frobenius, and we’ll call them Frobenius solutions.

It can be shown that the power series in a Frobenius solution of (eq:7.5.1) converges on some open interval , where . However, since may be complex for negative or undefined if , we’ll consider solutions defined for positive values of . Easy modifications of our results yield solutions defined for negative values of .

We’ll restrict our attention to the case where , , and are polynomials of degree not greater than two, so (eq:7.5.1) becomes

where , , and are real constants and . Most equations that arise in applications can be written this way. Some examples are
where we would multiply the last equation through by to put it in the form (eq:7.5.2). However, the method of Frobenius can be extended to the case where , , and are functions that can be represented by power series in on some interval that contains zero, and .

The next two theorems will enable us to develop systematic methods for finding Frobenius solutions of (eq:7.5.2).

We begin by showing that if is given by (eq:7.5.3) and , , and are constants, then where Differentiating (3) twice yields and Multiplying (eq:7.5.7) by and (eq:7.5.8) by yields and Therefore
which proves (eq:7.5.6).

Multiplying (eq:7.5.6) by yields

Multiplying (eq:7.5.6) by yields

To use these results, we rewrite as

From (eq:7.5.6) with , From (eq:7.5.9) with , From (eq:7.5.10) with , Therefore we can rewrite (eq:7.5.11) as
which implies (eq:7.5.4) with defined as in (eq:7.5.5).

If is determined by the recurrence relation (eq:7.5.12) then substituting into (eq:7.5.5) yields and for , so (eq:7.5.4) reduces to (eq:7.5.14). We omit the proof that the series (eq:7.5.13) converges on .

If for , then reduces to the Euler equation Theorem thmtype:7.4.3 shows that the solutions of this equation are determined by the zeros of the indicial polynomial Since (eq:7.5.14) implies that this is also true for the solutions of , we’ll also say that is the indicial polynomial of (eq:7.5.2), and that is the indicial equation of . We’ll consider only cases where the indicial equation has real roots and , with .

Since and are roots of , the indicial polynomial can be factored as Therefore which is nonzero if , since . Therefore the assumptions of Theorem thmtype:7.5.2 hold with , and (eq:7.5.14) implies that .

Now suppose isn’t an integer. From (eq:7.5.15), Hence, the assumptions of Theorem thmtype:7.5.2 hold with , and (eq:7.5.14) implies that . We leave the proof that is a fundamental set of solutions as an exercise.

It isn’t always possible to obtain explicit formulas for the coefficients in Frobenius solutions. However, we can always set up the recurrence relations and use them to compute as many coefficients as we want. The next example illustrates this.

Special Cases With Two Term Recurrence Relations

For , the recurrence relation (eq:7.5.12) of Theorem thmtype:7.5.2 involves the three coefficients , , and . We’ll now consider some special cases where (eq:7.5.12) reduces to a two term recurrence relation; that is, a relation involving only and or only and . This simplification often makes it possible to obtain explicit formulas for the coefficents of Frobenius solutions.

We first consider equations of the form with . For this equation, , so and the recurrence relations in Theorem thmtype:7.5.2 simplify to

We now consider equations of the form

with . For this equation, , so and the recurrence relations in Theorem thmtype:7.5.2 simplify to
Since , the last equation implies that if is odd, so the Frobenius solutions are of the form where

A Note on Technology

As we said at the end of Trench 7.2, if you’re interested in actually using series to compute numerical approximations to solutions of a differential equation, then whether or not there’s a simple closed form for the coefficents is essentially irrelevant; recursive computation is usually more efficient. Since it’s also laborious, we encourage you to write short programs to implement recurrence relations on a calculator or computer, even in exercises where this is not specifically required.

In practical use of the method of Frobenius when is a regular singular point, we’re interested in how well the functions approximate solutions to a given equation when is a zero of the indicial polynomial. In dealing with the corresponding problem for the case where is an ordinary point, we used numerical integration to solve the differential equation subject to initial conditions , and compared the result with values of the Taylor polynomial We can’t do that here, since in general we can’t prescribe arbitrary initial values for solutions of a differential equation at a singular point. Therefore, motivated by Theorem thmtype:7.5.2 (specifically, (eq:7.5.14)), we suggest the following procedure.

The multiplier on the right of (eq:7.5.27) eliminates the effects of small or large values of near , and of multiplication by an arbitrary constant.

To implement this procedure, you’ll have to write a computer program to calculate from the applicable recurrence relation, and to evaluate .

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)