Link to actual problem [11646] \[ \boxed {y+\left (y^{2} x +x -y\right ) y^{\prime }=0} \]
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 {x y -1}{x \,y^{2}+x -y}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y^{2}}{2}+\ln \left (x y-1\right )\right ] \\ \end{align*}