Link to actual problem [9570] \[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-\left (3 x +1\right ) y^{\prime }-\left (x^{2}-x \right ) y=0} \]
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}{\left (1+x \right )^{2}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{{\mathrm e}^{-2} \left (1+x \right )^{2} \operatorname {expIntegral}_{1}\left (-2 x -2\right )+2 \,{\mathrm e}^{2 x}}\right ] \\ \end{align*}