Link to actual problem [4118] \[ \boxed {\left (1+3 x \right ) {y^{\prime }}^{2}-3 \left (2+y\right ) y^{\prime }=-9} \]
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 &= \frac {1}{3}+\frac {y}{6}, \underline {\hspace {1.25 ex}}\eta &= 1\right ] \\ \left [R &= \frac {y^{2}}{12}-x +\frac {y}{3}, S \left (R \right ) &= y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 2 x -\frac {y}{3}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {3 x -y}{3 y^{2}}, S \left (R \right ) &= \ln \left (y\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x y -6 x +\frac {4}{3} y, \underline {\hspace {1.25 ex}}\eta &= y^{2}-6 x\right ] \\ \left [R &= -\frac {27 \left (-y^{2}-4 y+12 x \right )}{9 x^{2}-12 x y+4 y^{2}}, S \left (R \right ) &= -\frac {1}{y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x \,y^{2}-6 x^{2}-2 x y +\frac {1}{3} y^{2}+18 x -4 y, \underline {\hspace {1.25 ex}}\eta &= y^{3}-9 x y +18 x\right ] \\ \operatorname {FAIL} \\ \end{align*}