recursion relation, please help

You need to show some work. Let $y=\sum_{k=0}^\infty a_kx^k$ . Write y', y'', xy', xy'' as $\sum_{k=k_0}^\infty b_kx^k$ where $k_0=0$ or $k_0=1$ and $b_k$ is an expression of k and $a_i$ 's for some $i$ . The idea is to have the $k$ th term of each series to be some coefficient times $x^{k}$ (and not, say, $x^{k+1}$ or $x^{k-1}$ ). Then write the original equation in terms of series and see what equations on $a_k$ you get for k = 0 and for k > 0.

Feel free to post intermediate results for checking.