Link to actual problem [15056] \[ \boxed {\left (x^{2}+y^{2}+1\right ) y^{\prime }+y x=0} \]
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 {y \left (2 x^{2}+y^{2}+2\right )}{x^{2}+y^{2}+1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (2 x^{2}+y^{2}+2\right )}{4}+\frac {\ln \left (y\right )}{2}\right ] \\ \end{align*}