Link to actual problem [8624] \[ \boxed {\left (6 y^{2}-3 y x^{2}+1\right ) y^{\prime }-3 x y^{2}=-x} \]
type detected by program
{"exact"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
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 {1}{3 x^{2} y -6 y^{2}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {3 x^{2} y^{2}}{2}-2 y^{3}-y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {3 x^{2} y^{2}-4 y^{3}-x^{2}-2 y}{9 x^{2} y -18 y^{2}-3}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {3 \ln \left (-3 x^{2} y^{2}+4 y^{3}+x^{2}+2 y\right )}{2}\right ] \\ \end{align*}