Link to actual problem [7476] \[ \boxed {4 y^{\prime \prime } x^{2}+4 x^{5} y^{\prime }+\left (x^{8}+6 x^{4}+4\right ) y=0} \]
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 &= y \,{\mathrm e}^{\frac {x^{4}}{8}}, S \left (R \right ) &= -\frac {\ln \left (x \right )}{2}\right ] \\ \end{align*}