Link to actual problem [9208] \[ \boxed {y^{\prime }+\frac {-x y-y+x^{5} \sqrt {y^{2}+x^{2}}-x^{4} \sqrt {y^{2}+x^{2}}\, y}{x \left (x +1\right )}=0} \]
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 &= \frac {1+x}{x^{4}}, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (1+x \right ) y}{x^{5}}\right ] \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= \frac {x^{4}}{4}-\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (1+x \right )\right ] \\ \end{align*}