Link to actual problem [9702] \[ \boxed {y^{\prime \prime }+\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}+\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{2}-1\right )^{a -\frac {n}{2}} \operatorname {LegendreP}\left (v , n , x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right )^{-a} \left (x^{2}-1\right )^{\frac {n}{2}} y}{\operatorname {LegendreP}\left (v , n , x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{2}-1\right )^{a -\frac {n}{2}} \operatorname {LegendreQ}\left (v , n , x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right )^{-a} \left (x^{2}-1\right )^{\frac {n}{2}} y}{\operatorname {LegendreQ}\left (v , n , x\right )}\right ] \\ \end{align*}