Link to actual problem [5329] \[ \boxed {{y^{\prime }}^{2}-y^{\prime } x +y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {x}{2}\right ] \\ \left [R &= y-\frac {x^{2}}{4}, 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 {-x^{2}+4 y}}, S \left (R \right ) &= \frac {\left (-x^{2}+4 y\right )^{\frac {3}{2}} \left (\frac {4 y^{2}}{-x^{2}+4 y}-x^{2}\right )^{\frac {3}{2}}+\left (-x^{2}+4 y\right )^{\frac {3}{2}} x^{2} \sqrt {\frac {4 y^{2}}{-x^{2}+4 y}-x^{2}}+4 \sqrt {-x^{2}+4 y}\, \arctan \left (\frac {x}{\sqrt {\frac {4 y^{2}}{-x^{2}+4 y}-x^{2}}}\right ) y^{2} x -4 \sqrt {-x^{2}+4 y}\, \arctan \left (\frac {x}{\sqrt {\left (x +\frac {2 y}{\sqrt {-x^{2}+4 y}}\right ) \left (-x +\frac {2 y}{\sqrt {-x^{2}+4 y}}\right )}}\right ) y^{2} x -8 y^{3}}{8 x y^{3}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {\left (-x^{6} \left (x^{2}-4 y\right )^{3}\right )^{\frac {1}{4}} y}{x^{2} \left (x^{2}-4 y\right )}, S \left (R \right ) &= \int _{}^{y}\frac {1}{\textit {\_a} \left (\frac {\left (\frac {2 \sqrt {-x^{6} \left (x^{2}-4 y\right )^{3}}\, y^{2} \textit {\_a}}{x^{4} \left (x^{2}-4 y\right )^{2}}+\sqrt {-\frac {4 y^{4} \textit {\_a}^{2}}{x^{2} \left (x^{2}-4 y\right )}-\textit {\_a}^{4}}\right ) x^{4} \left (x^{2}-4 y\right )^{2}}{\sqrt {-x^{6} \left (x^{2}-4 y\right )^{3}}\, y^{2}}-2 \textit {\_a} \right )}d \textit {\_a}\right ] \\ \end{align*}