Link to actual problem [7061] \[ \boxed {\frac {{y^{\prime }}^{2}}{4}-x y^{\prime }+y=0} \]
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*} \\ \\ \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 [\underline {\hspace {1.25 ex}}\xi &= x^{2}-\frac {y}{2}, \underline {\hspace {1.25 ex}}\eta &= x y\right ] \\ \left [R &= \frac {y}{\sqrt {-x^{2}+y}}, S \left (R \right ) &= \frac {\left (-x^{2}+y\right )^{\frac {3}{2}} \left (\frac {y^{2}}{-x^{2}+y}-4 x^{2}\right )^{\frac {3}{2}}+4 \left (-x^{2}+y\right )^{\frac {3}{2}} x^{2} \sqrt {\frac {y^{2}}{-x^{2}+y}-4 x^{2}}-2 \sqrt {-x^{2}+y}\, \arctan \left (\frac {x \sqrt {4}}{\sqrt {\left (\frac {y}{\sqrt {-x^{2}+y}}+2 x \right ) \left (\frac {y}{\sqrt {-x^{2}+y}}-2 x \right )}}\right ) x y^{2}+2 \sqrt {-x^{2}+y}\, \arctan \left (\frac {2 x}{\sqrt {\frac {y^{2}}{-x^{2}+y}-4 x^{2}}}\right ) x y^{2}-y^{3}}{x y^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x \left (4 x^{2}-3 y \right )}{4}, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (2 x^{2}-y \right )}{2}\right ] \\ \left [R &= \frac {\left (-x^{6} \left (x^{2}-y\right )^{3}\right )^{\frac {1}{4}} y}{x^{2} \left (x^{2}-y\right )}, S \left (R \right ) &= \int _{}^{y}\frac {1}{\frac {\left (\frac {2 \sqrt {-x^{6} \left (x^{2}-y\right )^{3}}\, y^{2} \textit {\_a}}{x^{4} \left (x^{2}-y\right )^{2}}+2 \sqrt {-\frac {y^{4} \textit {\_a}^{2}}{x^{2} \left (x^{2}-y\right )}-4 \textit {\_a}^{4}}\right ) x^{4} \left (x^{2}-y\right )^{2} \textit {\_a}}{4 \sqrt {-x^{6} \left (x^{2}-y\right )^{3}}\, y^{2}}-\frac {\textit {\_a}^{2}}{2}}d \textit {\_a}\right ] \\ \end{align*}