Link to actual problem [8216] \[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y=0} \]
type detected by program
{"kovacic"}
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 &= \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}-\frac {1}{2}\right ], \left [-\frac {1}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}-\frac {1}{2}\right ], \left [-\frac {1}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (2 x +2\right )^{\frac {3}{2}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {5}}{2}+1, \frac {\sqrt {5}}{2}+1\right ], \left [\frac {5}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{2 \left (1+x \right ) \sqrt {2 x +2}\, \operatorname {hypergeom}\left (\left [-\frac {\sqrt {5}}{2}+1, \frac {\sqrt {5}}{2}+1\right ], \left [\frac {5}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}