Link to actual problem [8077] \[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y=0} \]
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 {\left (3 x +4\right ) \left (2 x -1\right )^{\frac {9}{2}}}{x^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{6} y}{\left (3 x +4\right ) \left (2 x -1\right )^{\frac {9}{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {231 x^{3}-198 x^{2}+66 x -8}{x^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{6} y}{231 x^{3}-198 x^{2}+66 x -8}\right ] \\ \end{align*}