Find power series solution for $y''-xy=0$ close to $x=0$. Find power series solution for $y''-xy=0$ close to $x=0$.
$\sum_{n=2}^\infty n(n-1)a_nx^{n-2}-x\sum_{n=0}^\infty a_nx^n=0$
Then,
$a_{n+2}(n+2)(n+1)-a_{n-1}=0 \implies a_{n+2}(n^2+3n+2)-a_{n-1}=0 \implies a_{n+3}=a_n\cdot \frac{1}{(n+3)(n+2)}$
I got two problems:
$(1)$ The answer is $a_{n+3}=(n+3)(n+2)a_n$ , where am I wrong ?
$(2)$ How can I find the general power series solution ?

If you look at the pattern (not very simple, I agree), there is a closed form for the coefficients
$$a_n=\frac{3^{-2 n/3}\,\, \Gamma \left(\frac{2}{3}\right)}{   \Gamma \left(\frac{n+2}{3}\right) \Gamma \left(\frac{n+3}{3}\right)}\left(a_0+a_1 \cos \left(\frac{2 \pi  n}{3}\right)+a_2\sin \left(\frac{2 \pi  n}{3}\right) \right)$$
