Link to actual problem [9455] \[ \boxed {y^{\prime \prime } x +\left (2 a \,x^{3}-1\right ) y^{\prime }+\left (a^{2} x^{3}+a \right ) x^{2} y=0} \]
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 &= -\frac {x}{a}, \underline {\hspace {1.25 ex}}\eta &= y \,x^{3}\right ] \\ \left [R &= y \,{\mathrm e}^{\frac {a \,x^{3}}{3}}, S \left (R \right ) &= -a \ln \left (x \right )\right ] \\ \end{align*}