Find two power series solutions of the given differential equation about the ordinary point x = 0. compare the series solutions with the solutions of the differential equation obtained using the method of section 4.3. try to explain any differences between the two forms of the solution. y'' − y' = 0

I don't know what method is referred to in "section 4.3", but I'll suppose it's reduction of order and use that to find the exact solution. Take , so that and we're left with the ODE linear in :



Now suppose has a power series expansion





Then the ODE can be written as







All the coefficients of the series vanish, and setting in the power series forms for and tell us that and , so we get the recurrence



We can solve explicitly for quite easily:



and so on. Continuing in this way we end up with



so that the solution to the ODE is



We also require the solution to satisfy , which we can do easily by adding and subtracting a constant as needed:

Welp life just is hard