Link to actual problem [9220] \[ \boxed {y^{\prime }-\frac {y a^{2} x +a +a^{2} x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a x y+1}{a^{2} x^{2} \left (a x y+1+a x \right )}=0} \]
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 (a x y +1\right )^{3}}{x^{2} \left (a x y +x a +1\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{2} \left (-\frac {1}{a x \left (a x y+1\right )}-\frac {1}{2 \left (a x y+1\right )^{2}}\right )\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}+2 a^{2} x y +x \,a^{2}+4 a x y +2 a +2\right )}{x \left (a x y +x a +1\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (2 y^{2} a^{2} x^{2}+2 y a^{2} x +x \,a^{2}+4 a x y+2 a +2\right )}{2 a^{2}}+\frac {\ln \left (a x y+1\right )}{a^{2}}\right ] \\ \end{align*}