Link to actual problem [4027] \[ \boxed {{y^{\prime }}^{2}-\left (2-x \right ) y^{\prime }-y=-1} \]
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 &= 1-\frac {x}{2}\right ] \\ \left [R &= y+\frac {x^{2}}{4}-x, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -1+\frac {x}{2}, \underline {\hspace {1.25 ex}}\eta &= y -1\right ] \\ \left [R &= \frac {y-1}{\left (-2+x \right )^{2}}, S \left (R \right ) &= 2 \ln \left (-2+x \right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {x^{2}-4 x +4 y}{4 x^{2}-16 x y+16 y^{2}-16 x +32 y+16}, S \left (R \right ) &= \int _{}^{y}\frac {1}{\frac {2 \textit {\_a} \left (\frac {\left (x^{2}-4 x +4 y\right ) \textit {\_a}}{x^{2}-4 x y+4 y^{2}-4 x +8 y+4}+\sqrt {\frac {\left (x^{2}-4 x +4 y\right ) \textit {\_a}^{2}}{x^{2}-4 x y+4 y^{2}-4 x +8 y+4}+\frac {\left (x^{2}-4 x +4 y\right ) \textit {\_a}}{x^{2}-4 x y+4 y^{2}-4 x +8 y+4}-\frac {x^{2}-4 x +4 y}{x^{2}-4 x y+4 y^{2}-4 x +8 y+4}-\textit {\_a} +1}+\frac {x^{2}-4 x +4 y}{x^{2}-4 x y+4 y^{2}-4 x +8 y+4}-1\right )}{\frac {x^{2}-4 x +4 y}{x^{2}-4 x y+4 y^{2}-4 x +8 y+4}-1}-2}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \\ \operatorname {FAIL} \\ \end{align*}