Link to actual problem [9212] \[ \boxed {y^{\prime }+\frac {-x y-y+\sqrt {y^{2}+x^{2}}\, x^{2}-x y \sqrt {y^{2}+x^{2}}}{x \left (x +1\right )}=0} \]
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 &= 1+\frac {1}{x}, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (1+x \right ) y}{x^{2}}\right ] \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= -\ln \left (1+x \right )+x\right ] \\ \end{align*}