Link to actual problem [9593] \[ \boxed {\left (x -1\right ) \left (x -2\right ) y^{\prime \prime }-\left (2 x -3\right ) y^{\prime }+y=0} \]
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 {5}{2}-\frac {\sqrt {5}}{2}, \frac {1}{2}-\frac {\sqrt {5}}{2}\right ], \left [-\sqrt {5}+1\right ], \frac {1}{-1+x}\right ) \left (-2+x \right )^{2} \left (-1+x \right )^{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{-\frac {\sqrt {5}}{2}} \sqrt {-1+x}\, y}{\operatorname {hypergeom}\left (\left [\frac {5}{2}-\frac {\sqrt {5}}{2}, \frac {1}{2}-\frac {\sqrt {5}}{2}\right ], \left [-\sqrt {5}+1\right ], \frac {1}{-1+x}\right ) \left (-2+x \right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [\frac {5}{2}+\frac {\sqrt {5}}{2}, \frac {1}{2}+\frac {\sqrt {5}}{2}\right ], \left [\sqrt {5}+1\right ], \frac {1}{-1+x}\right ) \left (-2+x \right )^{2} \left (-1+x \right )^{-\frac {1}{2}-\frac {\sqrt {5}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {-1+x}\, \left (-1+x \right )^{\frac {\sqrt {5}}{2}} y}{\operatorname {hypergeom}\left (\left [\frac {5}{2}+\frac {\sqrt {5}}{2}, \frac {1}{2}+\frac {\sqrt {5}}{2}\right ], \left [\sqrt {5}+1\right ], \frac {1}{-1+x}\right ) \left (-2+x \right )^{2}}\right ] \\ \end{align*}