Link to actual problem [15076] \[ \boxed {-y x^{2}+x^{2} \left (y-x \right ) y^{\prime }=-1} \]
type detected by program
{"exactWithIntegrationFactor"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{x -y}\right ] \\ \left [R &= x, S \left (R \right ) &= x y-\frac {y^{2}}{2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {2 x^{2} y -x \,y^{2}+2}{x \left (x -y \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (-2 x^{2} y+x y^{2}-2\right )}{2}\right ] \\ \end{align*}