Link to actual problem [8162] \[ \boxed {x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (3 x +2\right ) y=0} \]
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 +4\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-x^{3}-4 x^{2}\right ) {\mathrm e}^{x} \operatorname {expIntegral}_{1}\left (x \right )-2+x^{2}+3 x +\frac {2}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{-x^{3} {\mathrm e}^{x} \left (x +4\right ) \operatorname {expIntegral}_{1}\left (x \right )+x^{3}+3 x^{2}-2 x +2}\right ] \\ \end{align*}