Link to actual problem [9522] \[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (x -9\right ) y=0} \]
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}-8 x +20}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} y}{x^{2}-8 x +20}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-x} \left (x^{3}+9 x^{2}+36 x +60\right )}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} {\mathrm e}^{x} y}{x^{3}+9 x^{2}+36 x +60}\right ] \\ \end{align*}