I was wondering if the solution to the following differential equation belongs to a class of special functions. If not, is it exactly solvable?
\begin{equation} \frac{d^{2}y}{dx^{2}}+(\beta+\frac{1}{x}+\frac{\alpha}{x(x-1)})\frac{dy}{dx}-\frac{\kappa^{2}}{x(x-1)}y=0 \end{equation}
where $\beta$, $\alpha$ and $\kappa$ are real numbers.
With a change of variable I was trying to fit it in the Hypergeometric or Heun class of differential equations, but no success yet!
For this to have an answer, I slightly expand my comment:
This ODE is a particular case of the confluent Heun equation $$\frac{{d}^{2}w}{{dz}^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\epsilon% \right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0,$$ with $\left(\alpha,\gamma,\delta,\epsilon,q\right)_{\text{Heun}}=\left(0,1-\alpha,\alpha,\beta,\kappa^2\right)_{\text{OP}}$.