Eigenfunctions.

Question

I have the following ODE: $$y''-2xy'+2\alpha y=0$$ whose solution $y(x)$ may be recursively represented as: $$a_{n+2} = \frac{a_n(2n-2\alpha)}{(n+2)(n+1)}$$ I have found the eigenvalues to be $-2\alpha$, however I find the manner whereby the eigenfunctions are found to be rather perplexing. I'd sincerely appreciate an explanation. For instance, I know that for $\alpha=0$, $a_2=\frac{a_0(0 - 0)}{2}$, but why would that entail $y_0(x) = a_0$? I mean, how was that derived?