Link to actual problem [8452] \[ \boxed {y^{\prime } x -x \left (y-x \right ) \sqrt {x^{2}+y^{2}}-y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
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 &= \frac {1}{x}, \underline {\hspace {1.25 ex}}\eta &= \frac {y}{x^{2}}\right ] \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= \frac {x^{2}}{2}\right ] \\ \end{align*}