Link to actual problem [9049] \[ \boxed {y^{\prime }-\frac {-x^{2}+x +2+2 x^{3} \sqrt {x^{2}-4 x +4 y}}{2 \left (x +1\right )}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(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^{3}}, \underline {\hspace {1.25 ex}}\eta &= -\frac {\left (1+x \right ) \left (-2+x \right )}{2 x^{3}}\right ] \\ \left [R &= y+\frac {x^{2}}{4}-x, S \left (R \right ) &= \frac {x^{3}}{3}-\frac {x^{2}}{2}+x -\ln \left (1+x \right )\right ] \\ \end{align*}