This is actually a very simple problem, but I am going to type everything out in case I really overlooked something
I am trying to find a recurrence relation in the series solution I got. The ODE is $y'' + xy'- y = 0$.
$\begin{align} y'' + xy' - y & = \sum_{n=2}^{\infty} n(n - 1)a_nx^{n-2} + x\sum_{n=1}^{\infty} na_nx^{n-1} - \sum_{n=0}^{\infty} a_nx^n \\ & = \sum_{n=2}^{\infty} n(n - 1)a_nx^{n-2} + \sum_{n=1}^{\infty} na_nx^n - \sum_{n=0}^{\infty} a_nx^n \\ & = \sum_{n=0}^{\infty} (n+2)(n+1)a_nx^n + \sum_{n=0}^{\infty} na_nx^n - \sum_{n=0}^{\infty} a_nx^n\\ & = \sum_{n=0}^{\infty} x^n[(n+1)(n+2)a_{n+2} - a_n + na_n ] \\ & = 0 \end{align}$
The power series is 0 iff $(n+1)(n+2)a_{n+2} - a_n(1- n) = 0 \iff a_{n+2} = \dfrac{a_n (1 - n)}{(n+1)(n+2)}$
So testing out various values of n = 1, 2, 3... led me to
$a_2 = \dfrac{a_0}{2}$
$a_3 = 0$
$a_4 = \dfrac{-a_2}{12} = \dfrac{a_0}{24}$
$a_5 = 0$
$a_6 = -\dfrac{a_4}{10} = \dfrac{a_0}{240}$
Okay so clearly the odd terms all go to 0 and there will be no odd terms. I know that the even terms alternate back and forth in series (so I will get an alternating series). I have no idea how the denominator can jump from 2 to 24 and then to 240 and then to a stunning 240*56. I see no pattern in how the denominator jumps at all, it jumps too fast if you ask me