Link to actual problem [4028] \[ \boxed {{y^{\prime }}^{2}+\left (a +x \right ) y^{\prime }-y=0} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {a}{2}-\frac {x}{2}\right ] \\ \left [R &= y+\frac {x^{2}}{4}+\frac {x a}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\left (x +a \right )^{2}}, S \left (R \right ) &= \ln \left (x +a \right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {x^{2}+2 x a +a^{2}+4 y}{16 a^{4}+32 x \,a^{3}+16 a^{2} x^{2}+32 a^{2} y+32 a x y+16 y^{2}}, S \left (R \right ) &= \int _{}^{y}-\frac {1}{\textit {\_a} \left (3 a -\frac {-\frac {\left (x^{2}+2 x a +a^{2}+4 y\right ) a^{3}}{a^{4}+2 x \,a^{3}+a^{2} x^{2}+2 a^{2} y+2 a x y+y^{2}}-\frac {\left (x^{2}+2 x a +a^{2}+4 y\right ) a \textit {\_a}}{a^{4}+2 x \,a^{3}+a^{2} x^{2}+2 a^{2} y+2 a x y+y^{2}}+2 \sqrt {\textit {\_a} \left (\frac {a^{2} \left (x^{2}+2 x a +a^{2}+4 y\right )}{a^{4}+2 x \,a^{3}+a^{2} x^{2}+2 a^{2} y+2 a x y+y^{2}}+\frac {\left (x^{2}+2 x a +a^{2}+4 y\right ) \textit {\_a}}{4 a^{4}+8 x \,a^{3}+4 a^{2} x^{2}+8 a^{2} y+8 a x y+4 y^{2}}-1\right )}+a}{\frac {a^{2} \left (x^{2}+2 x a +a^{2}+4 y\right )}{a^{4}+2 x \,a^{3}+a^{2} x^{2}+2 a^{2} y+2 a x y+y^{2}}-1}\right )}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \\ \operatorname {FAIL} \\ \end{align*}