Link to actual problem [8088] \[ \boxed {3 x^{2} \left (x +3\right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y=0} \]
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}-36 x -243}{x^{\frac {2}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {2}{3}} y}{x^{2}-36 x -243}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {7 x +27}{x^{\frac {2}{3}} \left (x +3\right )^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {2}{3}} \left (x +3\right )^{\frac {1}{3}} y}{7 x +27}\right ] \\ \end{align*}