Link to actual problem [3948] \[ \boxed {\left (2-10 y^{3} x^{2}+3 y^{2}\right ) y^{\prime }-x \left (1+5 y^{4}\right )=0} \]
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}{10 x^{2} y^{3}-3 y^{2}-2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {5 y^{4} x^{2}}{2}-y^{3}-2 y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {5 x^{2} y^{4}-2 y^{3}+x^{2}-4 y}{50 x^{2} y^{3}-15 y^{2}-10}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {5 \ln \left (5 y^{4} x^{2}-2 y^{3}+x^{2}-4 y\right )}{2}\right ] \\ \end{align*}