We consider the utilization of power series to determine solutions to more general differential equations.

### Series Solutions Near an Ordinary Point II

In this section we continue to find series solutions of initial value problems

where , and are polynomials and , so is an ordinary point of (eq:7.3.1). However, here we consider cases where the differential equation in (eq:7.3.1) is not of the form so Theorem thmtype:7.2.2 does not apply, and the computation of the coefficients is more complicated. For the equations considered here it’s difficult or impossible to obtain an explicit formula for in terms of . Nevertheless, we can calculate as many coefficients as we wish. The next three examples illustrate this. Find the coefficients in the series solution of the initial value problem

Here
The zeros of have absolute value , so Theorem thmtype:7.2.2 implies that the series solution
converges to the solution of (eq:7.3.2) on . Since
Shifting indices so the general term in each series is a constant multiple of yields
where
Therefore is a solution of if and only if
From the initial conditions in (eq:7.3.2), and . Setting in (eq:7.3.3) yields
Setting in (eq:7.3.3) yields
We leave it to you to compute from (eq:7.3.3) and show that
We also leave it to you to verify numerically that the Taylor polynomials converge
to the solution of (eq:7.3.2) on .

Find the coefficients in the series solution
of the initial value problem

Since the desired series is in powers of we rewrite the differential equation in (eq:7.3.4) as ,
with
Since
Shifting indices so that the general term in each series is a constant multiple of yields
where
and
Therefore is a solution of if and only if
and
From the initial conditions in (eq:7.3.4), and . We leave it to you to compute with (eq:7.3.5) and (eq:7.3.6)
and show that the solution of (eq:7.3.4) is
We also leave it to you to verify numerically that the Taylor polynomials
converge to the solution of (eq:7.3.4) on the interval of convergence of the power series
solution.

Find the coefficients in the series solution of the initial value problem

Here
Since
Shifting indices so that the general term in each series is a constant multiple of
yields
where
and
Therefore is a solution of if and only if
and
From the initial conditions in (eq:7.3.7), and . We leave it to you to compute with (eq:7.3.8) and (eq:7.3.9)
and show that the solution of (eq:7.3.7) is
We also leave it to you to verify numerically that the Taylor polynomials
converge to the solution of (eq:7.3.9) on the interval of convergence of the power series
solution.

### Text Source

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