Link to actual problem [8284] \[ \boxed {y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (1+3 x \right )^{2}}\right ) y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= -x -\frac {1}{3}, \underline {\hspace {1.25 ex}}\eta &= x y\right ] \\ \left [R &= \frac {y \,{\mathrm e}^{x}}{\left (1+3 x \right )^{\frac {1}{3}}}, S \left (R \right ) &= -\ln \left (-3 x -1\right )\right ] \\ \end{align*}