Link to actual problem [9187] \[ \boxed {y^{\prime }-\frac {14 y x +12+2 x +y^{3} x^{3}+6 y^{2} x^{2}}{x^{2} \left (y x +2+x \right )}=0} \]
type detected by program
{"exactWithIntegrationFactor"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]
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 {\left (x y +2\right )^{3}}{x^{2} \left (x y +x +2\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{2} \left (-\frac {1}{x \left (x y+2\right )}-\frac {1}{2 \left (x y+2\right )^{2}}\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x y +2\right ) \left (2 x^{2} y^{2}+10 x y +x +12\right )}{2 x \left (x y +x +2\right )}\right ] \\ \\ \end{align*}