Link to actual problem [9715] \[ \boxed {y^{\prime \prime }+\frac {\left (-1+3 x \right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {\left (v \left (x -1\right ) \left (v +1\right )-a^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}=0} \]
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 &= x^{-\frac {v}{2}} \left (-1+x \right )^{-\frac {a}{2}} \operatorname {hypergeom}\left (\left [-\frac {v}{2}-\frac {a}{2}, \frac {1}{2}-\frac {v}{2}-\frac {a}{2}\right ], \left [\frac {1}{2}-v \right ], x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {v}{2}} \left (-1+x \right )^{\frac {a}{2}} y}{\operatorname {hypergeom}\left (\left [-\frac {v}{2}-\frac {a}{2}, \frac {1}{2}-\frac {v}{2}-\frac {a}{2}\right ], \left [\frac {1}{2}-v \right ], x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{2}+\frac {v}{2}} \left (-1+x \right )^{-\frac {a}{2}} \operatorname {hypergeom}\left (\left [1+\frac {v}{2}-\frac {a}{2}, \frac {1}{2}+\frac {v}{2}-\frac {a}{2}\right ], \left [\frac {3}{2}+v \right ], x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {v}{2}} \left (-1+x \right )^{\frac {a}{2}} y}{\sqrt {x}\, \operatorname {hypergeom}\left (\left [1+\frac {v}{2}-\frac {a}{2}, \frac {1}{2}+\frac {v}{2}-\frac {a}{2}\right ], \left [\frac {3}{2}+v \right ], x\right )}\right ] \\ \end{align*}