Link to actual problem [7849] \[ \boxed {y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_y_method_1"}
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^{3}}{3}}, S \left (R \right ) &= x\right ] \\ \end{align*}