In applying a Fourier-series type method to solve a PDE, I've encountered the following ode:
$$a\frac{d^2\phi}{dx^2}+bx\frac{d\phi}{dx}+\lambda\phi=0$$
where $a$ and $b$ are positive constants and $\lambda$ is unknown. This can be put in the standard Sturm-Liouville form
$$\frac{d}{dx}\left(e^{x^2/2}\frac{d\phi}{dx}\right)=-\lambda e^{x^2/2}\phi$$
with some straightforward change-of-variables. From this we can conclude the solutions will form an orthogonal basis and that this $\lambda$ is the eigenvalue. However, I can't find any known solutions for this weight function. For $w=e^{-x^2/2}$, we have the Hermite polynomials, but I cannot find any known solutions for $w=e^{x^2/2}$. Does anyone know the name of this class of solution, or any information on how to classify the behavior of the solutions $\phi$?
EDIT: I neglected to mention the domain for this problem is the finite interval $(0,1)$.
EDIT 2: I'm guessing the family of solutions will not be polynomials, since for $\lambda=0$ the solution is the error function. I will also add the solution is subject to a Robin condition at $x=1$ and a Nuemann condition at $x=0$, but these are optional. Only orthogonality on $(0,1)$ is required.