Link to actual problem [4385] \[ \boxed {\frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}}=0} \]
type detected by program
{"exact"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _exact]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= x^{2}+y^{2}, S \left (R \right ) &= -\arctan \left (\frac {x}{y}\right )\right ] \\ \end{align*}