Link to actual problem [3956] \[ \boxed {\left (4 x -x y^{3}-2 y^{4}\right ) y^{\prime }-\left (2+y^{3}\right ) y=0} \]
type detected by program
{"exactWithIntegrationFactor"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y^{3}}{x \,y^{3}+2 y^{4}-4 x}\right ] \\ \left [R &= x, S \left (R \right ) &= y^{2}+x y+\frac {2 x}{y^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (x \,y^{3}+y^{4}+2 x \right )}{x \,y^{3}+2 y^{4}-4 x}\right ] \\ \\ \end{align*}