Link to actual problem [4151] \[ \boxed {\left (-x^{2}+1\right ) {y^{\prime }}^{2}+2 y y^{\prime } x=-4 x^{2}} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
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 {x y}{2}, \underline {\hspace {1.25 ex}}\eta &= 2 x^{2}+y^{2}-4\right ] \\ \left [R &= \frac {y^{2}+4 x^{2}-4}{x^{4}}, S \left (R \right ) &= \int _{}^{x}\frac {2}{\sqrt {\frac {\left (y^{2}+4 x^{2}-4\right ) \textit {\_a}^{4}}{x^{4}}-4 \textit {\_a}^{2}+4}\, \textit {\_a}}d \textit {\_a}\right ] \\ \end{align*}