Link to actual problem [9654] \[ \boxed {y^{\prime \prime }-\frac {2 y^{\prime }}{x \left (x -2\right )}+\frac {y}{x^{2} \left (x -2\right )}=0} \]
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 {\sqrt {2}}{2}} \left (-2+x \right )^{2} \operatorname {hypergeom}\left (\left [1-\frac {\sqrt {2}}{2}, 2-\frac {\sqrt {2}}{2}\right ], \left [1-\sqrt {2}\right ], \frac {x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {\sqrt {2}}{2}} y}{\left (-2+x \right )^{2} \operatorname {hypergeom}\left (\left [1-\frac {\sqrt {2}}{2}, 2-\frac {\sqrt {2}}{2}\right ], \left [1-\sqrt {2}\right ], \frac {x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {\sqrt {2}}{2}} \left (-2+x \right )^{2} \operatorname {hypergeom}\left (\left [2+\frac {\sqrt {2}}{2}, 1+\frac {\sqrt {2}}{2}\right ], \left [1+\sqrt {2}\right ], \frac {x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {\sqrt {2}}{2}} y}{\left (-2+x \right )^{2} \operatorname {hypergeom}\left (\left [2+\frac {\sqrt {2}}{2}, 1+\frac {\sqrt {2}}{2}\right ], \left [1+\sqrt {2}\right ], \frac {x}{2}\right )}\right ] \\ \end{align*}