Link to actual problem [5103] \[ \boxed {-y^{2} x -\left (x +y x^{2}\right ) y^{\prime }=-x} \]
type detected by program
{"exactWithIntegrationFactor"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]
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 &= \frac {1}{\sqrt {\left (y -1\right ) \left (y +1\right )}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \sqrt {y^{2}-1}\, x\right ] \\ \end{align*}