Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x).
Note that this is entry-level calculus, so in my opinion there's gotta be something that makes this easier than it looks. Also note that $u_n(x)$ is the polynomial such that $\frac{d^n}{dx^n}e^\frac{x^2}{2} = u_n(x)e^\frac{x^2}{2}$.
What did I miss here?
Thank you for any hints.
Upon closer inspection during dinner it appears that $a_i = 0$ if $i$ is odd, and $$a_i = \frac{a_{i-2}(n-i+2)(n-i+1)}{i}$$ if i is even, where $a_0=1$.