Link to actual problem [10991] \[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+y c=0} \]
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 &= \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {a}{2}-\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}, -\frac {1}{2}-\frac {a}{2}+\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}\right ], \left [-\frac {a}{2}+\frac {b}{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 {a}{2}-\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}, -\frac {1}{2}-\frac {a}{2}+\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}\right ], \left [-\frac {a}{2}+\frac {b}{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 (\frac {x}{2}+\frac {1}{2}\right )^{1+\frac {a}{2}-\frac {b}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}-\frac {b}{2}, \frac {1}{2}+\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}-\frac {b}{2}\right ], \left [2+\frac {a}{2}-\frac {b}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 \left (\frac {x}{2}+\frac {1}{2}\right )^{-\frac {a}{2}} \left (\frac {x}{2}+\frac {1}{2}\right )^{\frac {b}{2}} y}{\left (1+x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}-\frac {b}{2}, \frac {1}{2}+\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}-\frac {b}{2}\right ], \left [2+\frac {a}{2}-\frac {b}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}