Link to actual problem [6880] \[ \boxed {x^{2} {y^{\prime }}^{2}-\left (2 y x +1\right ) y^{\prime }+y^{2}=-1} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x^{2}, \underline {\hspace {1.25 ex}}\eta &= x y +\frac {1}{2}\right ] \\ \left [R &= \frac {1+4 x y}{4 x^{2}}, S \left (R \right ) &= -\frac {1}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x y +\frac {1}{2}, \underline {\hspace {1.25 ex}}\eta &= y^{2}+1\right ] \\ \left [R &= \frac {\sqrt {-2 y^{2}-2}}{2 x -y}, S \left (R \right ) &= \arctan \left (y\right )\right ] \\ \end{align*}