Link to actual problem [9313] \[ \boxed {y^{\prime }-\frac {x^{3} y^{3}+6 x^{2} y^{2}+12 x y+8+2 x}{x^{3}}=0} \]
type detected by program
{"abelFirstKind", "exactWithIntegrationFactor"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]
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^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {x^{2}}{2 \left (x y+2\right )^{2}}\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}+8 x y +x +8\right )}{2 x^{2}}\right ] \\ \\ \end{align*}