Link to actual problem [9665] \[ \boxed {y^{\prime \prime }-\frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (x -2\right )}-\frac {y}{3 x^{2} \left (x -2\right )}=0} \]
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 &= \left (-2+x \right )^{\frac {17}{6}} x^{\frac {1}{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (-2+x \right )^{\frac {17}{6}} x^{\frac {1}{6}}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{18 x^{3}-102 x^{2}+187 x}\right ] \\ \end{align*}