Link to actual problem [9181] \[ \boxed {y^{\prime }-\sqrt {x^{2}+2 x +1-4 y}-x^{2} \sqrt {x^{2}+2 x +1-4 y}-\sqrt {x^{2}+2 x +1-4 y}\, x^{3}=\frac {x}{2}+\frac {1}{2}} \]
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^{3}+x^{2}+1}, \underline {\hspace {1.25 ex}}\eta &= \frac {1+x}{2 x^{3}+2 x^{2}+2}\right ] \\ \left [R &= y-\frac {x^{2}}{4}-\frac {x}{2}, S \left (R \right ) &= \frac {x \left (3 x^{3}+4 x^{2}+12\right )}{12}\right ] \\ \end{align*}