Link to actual problem [4033] \[ \boxed {{y^{\prime }}^{2}-2 x y^{\prime }+2 y=0} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _Clairaut]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= y-\frac {x^{2}}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{x^{2}}, S \left (R \right ) &= 2 \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {2 y-x^{2}}}, S \left (R \right ) &= -\frac {\left (2 y-x^{2}\right )^{\frac {3}{2}} \left (\frac {y^{2}}{2 y-x^{2}}-x^{2}\right )^{\frac {3}{2}}+\left (2 y-x^{2}\right )^{\frac {3}{2}} x^{2} \sqrt {\frac {y^{2}}{2 y-x^{2}}-x^{2}}+\sqrt {2 y-x^{2}}\, \arctan \left (\frac {x}{\sqrt {\frac {y^{2}}{2 y-x^{2}}-x^{2}}}\right ) y^{2} x -\sqrt {2 y-x^{2}}\, \arctan \left (\frac {x}{\sqrt {\left (\frac {y}{\sqrt {2 y-x^{2}}}+x \right ) \left (\frac {y}{\sqrt {2 y-x^{2}}}-x \right )}}\right ) y^{2} x +y^{3}}{y^{3} x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x \left (2 x^{2}-3 y \right )}{2}, \underline {\hspace {1.25 ex}}\eta &= y \left (x^{2}-y \right )\right ] \\ \left [R &= \frac {\left (-x^{6} \left (x^{2}-2 y\right )^{3}\right )^{\frac {1}{4}} y}{x^{2} \left (x^{2}-2 y\right )}, S \left (R \right ) &= \int _{}^{y}\frac {1}{\textit {\_a} \left (\frac {\left (\frac {\sqrt {-x^{6} \left (x^{2}-2 y\right )^{3}}\, y^{2} \textit {\_a}}{x^{4} \left (x^{2}-2 y\right )^{2}}+\sqrt {-\frac {y^{4} \textit {\_a}^{2}}{x^{2} \left (x^{2}-2 y\right )}-\textit {\_a}^{4}}\right ) x^{4} \left (x^{2}-2 y\right )^{2}}{\sqrt {-x^{6} \left (x^{2}-2 y\right )^{3}}\, y^{2}}-\textit {\_a} \right )}d \textit {\_a}\right ] \\ \end{align*}