Link to actual problem [8148] \[ \boxed {2 x^{2} y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+2 \left (4 x -1\right ) y=0} \]
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 &= \frac {x^{2} \left (4 x^{2}-36 x +63\right )}{63}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\frac {4}{63} x^{4}-\frac {4}{7} x^{3}+x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {hypergeom}\left (\left [-\frac {9}{2}\right ], \left [-\frac {3}{2}\right ], x\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\operatorname {hypergeom}\left (\left [-\frac {9}{2}\right ], \left [-\frac {3}{2}\right ], x\right )}\right ] \\ \end{align*}