Link to actual problem [9314] \[ \boxed {y^{\prime }-\frac {y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 y a x +1+a^{2} x}{x^{3} a^{3}}=0} \]
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 (a x y +1\right )^{3}}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {x^{2}}{2 \left (a x y+1\right )^{2} a}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (a x y +1\right ) \left (2 a^{2} x^{2} y^{2}+x \,a^{2}+4 a x y +2\right )}{2 x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (2 y^{2} a^{2} x^{2}+x \,a^{2}+4 a x y+2\right )}{a^{3}}+\frac {2 \ln \left (a x y+1\right )}{a^{3}}\right ] \\ \end{align*}