General solution in the form of a power series? HELP
y'' - 3xy' - y = 0
The question asks me to find the general solution of the equation in the form of a power series around x = 0.
Ive been at this for a good while now and I cant seem to finish it.
I end up with a series split into a0's and a1's, the questions ive done before this all had the numerator equal to one so turning the result into a power series was easier. Im left with horrible numerators that i cant find a pattern to transform it into a power series. Im dont know how to write the maths symbols, heres my best effort, this is what i end up with
y = a0 + a1x + a0/2!x2 + 4a1/3!x3 + 7a0/4!x4 + 40a1/5!x5 + 91a0/6!x6 + ........
Have I gone wrong somewhere?
I think im on the right track, its just the numerator thats ruining my life currently, How to turn to power series? Any help would be greatly appreciated.