Link to actual problem [10984] \[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [n +1, -n -\beta -\alpha \right ], \left [1-\beta \right ], \frac {x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [n +1, -n -\beta -\alpha \right ], \left [1-\beta \right ], \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\frac {x}{2}+\frac {1}{2}\right )^{\beta } \operatorname {hypergeom}\left (\left [-n -\alpha , n +\beta +1\right ], \left [\beta +1\right ], \frac {x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {x}{2}+\frac {1}{2}\right )^{-\beta } y}{\operatorname {hypergeom}\left (\left [-n -\alpha , n +\beta +1\right ], \left [\beta +1\right ], \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}