I am trying to solve the following differential equation:

\begin{equation} (1-x^{2})y''+\{\alpha x+1 \}y'+\{\beta^{2}x^{2}-\beta{x}+\gamma\}y=0 \end{equation}

where $y=y(x)$. $y'$ and $y''$ are the first and second derivatives respectively. $\alpha$, $\beta$ and $\gamma$ are constants.

There are various second order differential equations that are very similar looking to this equation. For instance Mathieu equation, Legendre equation and generalized Hypergeometric equation, but not quite the same. I am guessing that a change of variable can transform the equation into a known form. However, no success yet!

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    $\begingroup$ it is a Heun equation the solution containes non elementary functions $\endgroup$ – Dr. Sonnhard Graubner Jul 10 '15 at 16:57

For the equation \begin{equation} (1-x^{2})y''+\{\alpha x+1 \}y'+\{\beta^{2}x^{2}-\beta{x}+\gamma\}y=0 \end{equation} let $x = 2t -1$ to obtain the form \begin{align} t(t-1) y'' + \left( \frac{\alpha -1}{2} - \alpha t \right) y' + [ -(\gamma - \beta + \beta^{2}) + 2 \beta (t - 1) - 4 \beta^{2} \, t(t-1) ] y =0. \end{align} This equation is in the form of the Generalized Spheroidal Wave Equation, which is in the Confluent Heun equation class of differential equations. Some references can be obtained from equation (4) of Bartolomeu D B Figueiredo 's paper.


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