Link to actual problem [9012] \[ \boxed {y^{\prime }-\frac {x^{2} \left (x +1+2 x \sqrt {x^{3}-6 y}\right )}{2 \left (x +1\right )}=0} \]
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 {1}{2}+\frac {1}{2 x}\right ] \\ \left [R &= y-\frac {x^{3}}{6}, S \left (R \right ) &= \frac {x^{3}}{3}-\frac {x^{2}}{2}+x -\ln \left (1+x \right )\right ] \\ \end{align*}