I am trying to solve y''+(3-1/x^2)y=0. Is it possible to solve by separation? Or will this require a series solution?
If by "separation" you mean writing it as $\displaystyle \frac{d^2y}{dx^2}= (\frac{1}{x^2}- 3)y$ and then to
$\displaystyle \frac{d^2y}{y}= (\frac{1}{x^2}- 3)dx^2$, then, no, you cannot "separate" a second order derivative like you can a first order derivative. The "differentials" dy and dx are defined in Calculus, but such things as "$\displaystyle d^2y$" and "$\displaystyle dx^2$" are NOT defined.