Link to actual problem [9238] \[ \boxed {y^{\prime }-\frac {a^{2} x +a^{3} x^{3}+y^{2} a^{3} x^{3}+2 y a^{2} x^{2}+a x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a x y+1}{a^{3} x^{3}}=0} \]
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 &= \frac {a^{3} x^{3} y^{3}+a^{3} x^{3} y^{2}+a^{3} x^{3}+3 a^{2} x^{2} y^{2}+2 a^{2} x^{2} y +3 a x y +x a +1}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {x^{3}}{1+a^{3} \left (\textit {\_a}^{3}+\textit {\_a}^{2}+1\right ) x^{3}+3 a^{2} \left (\textit {\_a} +\frac {2}{3}\right ) \textit {\_a} \,x^{2}+a \left (3 \textit {\_a} +1\right ) x}d \textit {\_a}\right ] \\ \end{align*}