Assume that the differential equation

has a power-series solution and find a formula for the coefficient .

First, we have

Therefore, we have

Equating like powers of we obtain the recursive relation

By induction, we then have

Therefore,

It should be alpha^n/n!, at least. But how did you get that a_0 = 1?

To me it seems like it can be any constant so the solution is actually a0*alpha^n/n!.