Link to actual problem [9656] \[ \boxed {y^{\prime \prime }+\frac {\left (\left (\beta +\alpha +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (x -1\right ) \left (x -a \right )}+\frac {\left (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (x -a \right )}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunG}\left (a , q , \alpha , \beta , \gamma , \delta , x\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\gamma } y}{x \operatorname {HeunG}\left (a , \gamma ^{2}+\left (\left (1-a \right ) \delta -\alpha -\beta -2\right ) \gamma +\delta \left (a -1\right )+\alpha +\beta +q +1, \beta +1-\gamma , \alpha +1-\gamma , -\gamma +2, \delta , x\right )}\right ] \\ \end{align*}