Link to actual problem [3862] \[ \boxed {\left (x^{3}+2 y-y^{2}\right ) y^{\prime }+3 y x^{2}=0} \]
type detected by program
{"exact", "differentialType"}
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}{x^{3}-y^{2}+2 y}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{3} y-\frac {y^{3}}{3}+y^{2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (3 x^{3}-y^{2}+3 y \right )}{3 x^{3}-3 y^{2}+6 y}\right ] \\ \\ \end{align*}