# Question Regarding a Second Order Ordinary Differential Equation

I was wondering if the solution to the following differential equation belongs to a class of special functions. If not, is it exactly solvable?

$$\frac{d^{2}y}{dx^{2}}+(\beta+\frac{1}{x}+\frac{\alpha}{x(x-1)})\frac{dy}{dx}-\frac{\kappa^{2}}{x(x-1)}y=0$$

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!

• How come? if you check the requrality of this D.E. we will find that it is irreqular. so cannot use series solution near 0 nor 1. This means the discontinuity points are non-removable so cannot have analytic solution. Commented Aug 24, 2015 at 16:52
• Nope! $x=0$ and $x=1$ are both regular singular points.
– Ben
Commented Aug 24, 2015 at 16:55
• Yes you are right... sorry Commented Aug 24, 2015 at 16:58
• This is the confluent Heun equation. Commented Aug 24, 2015 at 21:21
• @Startwearingpurple: What form of CHE are you referring to?
– Ben
Commented Aug 24, 2015 at 21:34

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 $\left(\alpha,\gamma,\delta,\epsilon,q\right)_{\text{Heun}}=\left(0,1-\alpha,\alpha,\beta,\kappa^2\right)_{\text{OP}}$.