Link to actual problem [8151] \[ \boxed {x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (-x +4\right ) y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} y}{x^{2} \left (x^{2}+4 x +2\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left ({\mathrm e}^{x} \left (x^{2}+4 x +2\right ) \operatorname {expIntegral}_{1}\left (x \right )-x -3\right ) x^{2}}\right ] \\ \end{align*}