I am trying to solve for the Green's Function of the ODE
$$y''(x)-(1+x^2)y(x) = f(x), \hspace{.5cm} -\infty < x < \infty$$ with boundary conditions $y \to 0$ as $\left| x \right| \to \infty$. All methods I've used in the past to solve problems like these involved finding the solution to the homogeneous problem, but there doesn't appear to be a solution in terms of elementary functions. If we're just considering the homogeneous problem, I found the series solution $$y(x) = \sum_{n=0}^{\infty} a_nx^n,$$ where the coefficients satisfy $$a_{n+2} = \frac{a_n + a_{n-2}}{(n+2)(n+1)}, $$ and $a_0, a_1$ are arbitrary. From this, I was able to find \begin{align*} a_{2n} &= \frac{(2n-1)!!}{(2n)!}, \hspace{.2cm} n > 0 \\ \end{align*} which can be verified via induction. I haven't had the same luck with $a_{2n-1}$ though. Even if I were able to solve for these coefficients, would my series solution be useful in finding the Green's function? I'm not sure how applying the boundary conditions would reduce the complexity of the problem. Thanks in advance.