Consider the differential equation
with initial conditions when . Assume this differential equation has a power-series solution and compute the first four nonzero terms of the expansion.
be the power-series solution of the differential equation. Then we must have
From the initial conditions we know . Then, equating like powers of we can solve for the first four nonzero terms in the power series expansion:
(Note: I think the solution in the back of Apostol is wrong on this. Apostol has , , and . I’m going to mark this as errata until someone convinces me Apostol is actually correct.)
Therefore, we have