Link to actual problem [2489] \[ \boxed {y \left (2 x^{2} y^{2}+1\right ) y^{\prime }+x \left (y^{4}+1\right )=0} \]
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}{y \left (2 x^{2} y^{2}+1\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x^{2} y^{2}+1\right )^{2}}{8 x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{2} y^{4}+x^{2}+y^{2}}{y \left (2 x^{2} y^{2}+1\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (y^{4} x^{2}+x^{2}+y^{2}\right )}{2}\right ] \\ \end{align*}