Link to actual problem [594] \[ \boxed {y^{\prime }-\frac {x}{x^{2}+y+y^{3}}=0} \]
type detected by program
{"exactWithIntegrationFactor"}
type detected by Maple
[_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 {4 y^{3}+4 x^{2}+6 y^{2}+10 y +5}{4 y^{3}+4 x^{2}+4 y}\right ] \\ \left [R &= x, S \left (R \right ) &= y-\frac {\ln \left (4 y^{3}+4 x^{2}+6 y^{2}+10 y+5\right )}{2}\right ] \\ \end{align*}