Link to actual problem [9725] \[ \boxed {y^{\prime \prime }+\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}+\frac {\left (x^{2}-1-\left (1+2 v \right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
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 )^{-v -\frac {1}{2}} \operatorname {hypergeom}\left (\left [-v , -v \right ], \left [-2 v \right ], -x^{2}+1\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right )^{v} \sqrt {x^{2}-1}\, y}{\operatorname {hypergeom}\left (\left [-v , -v \right ], \left [-2 v \right ], -x^{2}+1\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 )^{v +\frac {1}{2}} \operatorname {hypergeom}\left (\left [v +1, v +1\right ], \left [2 v +2\right ], -x^{2}+1\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right )^{-v} y}{\sqrt {x^{2}-1}\, \operatorname {hypergeom}\left (\left [v +1, v +1\right ], \left [2 v +2\right ], -x^{2}+1\right )}\right ] \\ \end{align*}