Link to actual problem [9701] \[ \boxed {y^{\prime \prime }-\frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}+\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
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 {\left (x^{2}-1\right )^{-a} y}{\operatorname {LegendreP}\left (v , x\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right )^{-a} y}{\operatorname {LegendreQ}\left (v , x\right )}\right ] \\ \end{align*}