We conclude our study of the method of Frobenius for finding series solutions of linear second order differential equations, considering the case where the indicial equation has distinct real roots that differ by an integer.
The Method of Frobenius III
In Trench 7.5 and 7.6 we discussed methods for finding Frobenius solutions of a homogeneous linear second order equation near a regular singular point in the case where the indicial equation has a repeated root or distinct real roots that don’t differ by an integer. In this section we consider the case where the indicial equation has distinct real roots that differ by an integer. We’ll limit our discussion to equations that can be written as
or where the roots of the indicial equation differ by a positive integer.We begin with a theorem that provides a fundamental set of solutions of equations of the form (eq:7.7.1).
- Proof
- Theorem thmtype:7.5.3 implies that . We’ll now show that . Since is a linear operator,
this is equivalent to showing that
To verify this, we’ll show that
and
This will imply that , since substituting (eq:7.7.7) and (eq:7.7.8) into (eq:7.7.6) and using (eq:7.7.4) yields
We’ll prove (eq:7.7.8) first. From Theorem thmtype:7.6.1, Setting and recalling that and yields
Since and are the roots of the indicial equation, the indicial polynomial can be written as Differentiating this yields Therefore , so (eq:7.7.9) implies (eq:7.7.8).Before proving (eq:7.7.7), we first note is well defined by (eq:7.7.3) for , since for these values of . However, we can’t define for with (eq:7.7.3), since . For convenience, we define for . Then, from Theorem thmtype:7.5.1,
where and If , then (eq:7.7.3) implies that . If , then because . Therefore (eq:7.7.10) reduces to Since and , this implies (eq:7.7.7).We leave the proof that is a fundamental set as an exercise.
To compute the coefficients and in , we set in (eq:7.7.13) and apply the resulting recurrence formula for , ; thus,
We use logarithmic differentiation to obtain obtain . From (eq:7.7.14), Therefore Differentiating with respect to yields Therefore Setting here and recalling (eq:7.7.15) yields
Since we can rewrite (eq:7.7.17) asSubstituting this into (eq:7.7.16) yields
If in (eq:7.7.4), there’s no need to compute in the formula (eq:7.7.5) for . Therefore it’s best to compute before computing . This is illustrated in the next example.
To compute the coefficients in , we set in (eq:7.7.20) and apply the resulting recurrence formula for , , , ; thus,
We now consider equations of the form where the roots of the indicial equation are real and differ by an even integer. The proof of the next theorem is similar to the proof of Theorem thmtype:7.7.1
To compute the coefficients , , and in , we set in (eq:7.7.26) and apply the resulting recurrence formula for , , ; thus,
This yields Substituting , , , and into (eq:7.7.23) yields . Therefore, from (eq:7.7.24),
To obtain we use logarithmic differentiation. From (eq:7.7.27), Therefore Differentiating with respect to yields Therefore Setting here and recalling (eq:7.7.28) yields
Since we can rewrite (eq:7.7.30) as Substituting this into (eq:7.7.29) yieldsTo compute the coefficients , , and in , we set in (eq:7.7.33) and apply the resulting recurrence formula for , ; thus,
Text Source
Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)