Link to actual problem [9226] \[ \boxed {y^{\prime }-\frac {6 x +x^{3}+x^{3} y^{2}+4 x^{2} y+y^{3} x^{3}+6 y^{2} x^{2}+12 x y+8}{x^{3}}=0} \]
type detected by program
{"abelFirstKind"}
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 &= 1+\frac {\left (x y +x +2\right ) \left (x y +2\right )^{2}}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {x^{3}}{8+\left (\textit {\_a}^{3}+\textit {\_a}^{2}+1\right ) x^{3}+\left (6 \textit {\_a}^{2}+4 \textit {\_a} \right ) x^{2}+\left (12 \textit {\_a} +4\right ) x}d \textit {\_a}\right ] \\ \end{align*}