Link to actual problem [4381] \[ \boxed {y^{\prime } y+\frac {x y^{\prime }}{x^{2}+y^{2}}-\frac {y}{x^{2}+y^{2}}=-x} \]
type detected by program
{"exact"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _exact, _rational]
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 {x^{2}+y^{2}}{x^{2} y +y^{3}+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y^{2}}{2}+\arctan \left (\frac {y}{x}\right )\right ] \\ \end{align*}