Link to actual problem [4099] \[ \boxed {x {y^{\prime }}^{2}+\left (-y+x \right ) y^{\prime }-y=-1} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]
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 {3 x}{2}-\frac {y}{2}, \underline {\hspace {1.25 ex}}\eta &= -1+x\right ] \\ \left [R &= -\frac {2 x +y+1}{x^{2}+2 x y+y^{2}+4 x +4 y+4}, S \left (R \right ) &= -\ln \left (-1+x \right )-2 \,\operatorname {arctanh}\left (\sqrt {\frac {9 x^{2}+6 x y+y^{2}}{x^{2}+2 x y+y^{2}+4 x +4 y+4}}\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{2}-\frac {y}{2}, \underline {\hspace {1.25 ex}}\eta &= y -1\right ] \\ \left [R &= \frac {x +y-2}{\sqrt {y-1}}, S \left (R \right ) &= \ln \left (y-1\right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {4 y^{2}+8 x y+4 x^{2}-16 x}{x^{2}-2 x y+y^{2}-4 x +4 y+4}, S \left (R \right ) &= \int _{}^{y}\frac {1}{\frac {\textit {\_a} \left (\frac {4 \left (y^{2}+2 x y+x^{2}-4 x \right ) \textit {\_a}}{x^{2}-2 x y+y^{2}-4 x +4 y+4}+4 \sqrt {\frac {4 \left (y^{2}+2 x y+x^{2}-4 x \right ) \textit {\_a}^{2}}{x^{2}-2 x y+y^{2}-4 x +4 y+4}+\frac {4 \left (y^{2}+2 x y+x^{2}-4 x \right ) \textit {\_a}}{x^{2}-2 x y+y^{2}-4 x +4 y+4}-\frac {4 \left (y^{2}+2 x y+x^{2}-4 x \right )}{x^{2}-2 x y+y^{2}-4 x +4 y+4}-4 \textit {\_a} +4}+\frac {8 y^{2}+16 x y+8 x^{2}-32 x}{x^{2}-2 x y+y^{2}-4 x +4 y+4}+4 \textit {\_a} -8\right )}{\frac {4 y^{2}+8 x y+4 x^{2}-16 x}{x^{2}-2 x y+y^{2}-4 x +4 y+4}-4}+\textit {\_a}^{2}-2}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4}-4 x^{3}-8 x^{2} y-4 x y^{2}}{x^{4}}, S \left (R \right ) &= -\frac {\sqrt {\frac {x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4}-4 x^{3}-8 x^{2} y-4 x y^{2}+4 x^{2}}{x^{2}}}}{4 x}\right ] \\ \end{align*}