Link to actual problem [8181] \[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (3 x -1\right ) y=0} \]
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 \left (x -3\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{2}-3 x \right ) {\mathrm e}^{-x} \operatorname {expIntegral}_{1}\left (-x \right )-2+x -\frac {1}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{x} y}{x^{2} \left (x -3\right ) \operatorname {expIntegral}_{1}\left (-x \right )+\left (x^{2}-2 x -1\right ) {\mathrm e}^{x}}\right ] \\ \end{align*}