Link to actual problem [7958] \[ \boxed {\left (x +2\right ) y^{\prime \prime }+y^{\prime } x +3 y=0} \]
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 (x^{2}-6 x +4\right ) \left (2+x \right )^{3}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{{\mathrm e}^{-2} \left (x^{2}-6 x +4\right ) \left (2+x \right )^{3} \operatorname {Ei}_{1}\left (-2-x \right )+\left (x^{4}-x^{3}-18 x^{2}-22 x +8\right ) {\mathrm e}^{x}}\right ] \\ \end{align*}