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.

Let

be the power series solution of the differential equation. Then we must have

From the initial condition when we know . Therefore, equating like powers of we have

(**Note:** The book gives the value . I think the answer we have above is correct. I’m marking this as errata unless someone convinces me that Apostol is correct.)

Therefore, we have