Link to actual problem [9652] \[ \boxed {y^{\prime \prime }+\frac {\left (\left (a +1+b \right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (x -1\right )}+\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}=0} \]
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 {x^{-\alpha } \left (-1+x \right )^{a} \left (-1+x \right )^{\alpha } \left (-1+x \right )^{b} \left (-1+x \right )^{\beta } y}{\left (-1+x \right ) \operatorname {hypergeom}\left (\left [1-b -\beta , 1-a -\beta \right ], \left [1+\alpha -\beta \right ], x\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\beta } \left (-1+x \right )^{a} \left (-1+x \right )^{\alpha } \left (-1+x \right )^{b} \left (-1+x \right )^{\beta } y}{\left (-1+x \right ) \operatorname {hypergeom}\left (\left [1-\alpha -b , 1-a -\alpha \right ], \left [1-\alpha +\beta \right ], x\right )}\right ] \\ \end{align*}