Link to actual problem [4098] \[ \boxed {x {y^{\prime }}^{2}+\left (a -y\right ) y^{\prime }=-b} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= -\frac {2 a y+4 b x -y^{2}}{4 b}, S \left (R \right ) &= \frac {y}{2 b}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {-y+a}{\sqrt {x}}, S \left (R \right ) &= \frac {\ln \left (x \right )}{2}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y^{2}-2 a y+a^{2}-4 b x}{\left (a^{4}-2 y a^{3}-2 a^{2} b x +y^{2} a^{2}+2 a b x y+b^{2} x^{2}\right ) b}, S \left (R \right ) &= \int _{}^{y}\frac {1}{-9 a^{2}+\frac {2 \left (\frac {3 \left (y^{2}-2 a y+a^{2}-4 b x \right ) a^{2}}{a^{4}-2 y a^{3}-2 a^{2} b x +y^{2} a^{2}+2 a b x y+b^{2} x^{2}}+\frac {\left (y^{2}-2 a y+a^{2}-4 b x \right ) a \textit {\_a}}{a^{4}-2 y a^{3}-2 a^{2} b x +y^{2} a^{2}+2 a b x y+b^{2} x^{2}}+\sqrt {\frac {\left (y^{2}-2 a y+a^{2}-4 b x \right ) \textit {\_a}^{2}}{a^{4}-2 y a^{3}-2 a^{2} b x +y^{2} a^{2}+2 a b x y+b^{2} x^{2}}+\frac {2 \left (y^{2}-2 a y+a^{2}-4 b x \right ) a \textit {\_a}}{a^{4}-2 y a^{3}-2 a^{2} b x +y^{2} a^{2}+2 a b x y+b^{2} x^{2}}-\frac {3 \left (y^{2}-2 a y+a^{2}-4 b x \right ) a^{2}}{a^{4}-2 y a^{3}-2 a^{2} b x +y^{2} a^{2}+2 a b x y+b^{2} x^{2}}+4}+2\right ) \left (a^{4}-2 y a^{3}-2 a^{2} b x +y^{2} a^{2}+2 a b x y+b^{2} x^{2}\right )}{y^{2}-2 a y+a^{2}-4 b x}+\textit {\_a}^{2}}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \\ \operatorname {FAIL} \\ \end{align*}