Find the power series solutions of the equation:
(2x^{2})y'' + (3x)y' - (x^{2} + 1)y = 0
Any help would gladly be appreciated.
Start by writing $\displaystyle \displaystyle \begin{align*} y = \sum_{ k = 0 }^{\infty} c_k \, x^k \end{align*}$, which means $\displaystyle \displaystyle \begin{align*} y' = k\, c_k \, x^{k-1} \end{align*}$ and $\displaystyle \displaystyle \begin{align*} y'' = k\left( k - 1 \right) c_k \, x^{k - 2} \end{align*}$.
Substitute these in and perform some algebraic manipulation. What can you come up with?