Link to actual problem [6884] \[ \boxed {{y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{\prime } x -y=0} \]
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 &= \frac {2 x}{3}-\frac {2}{9}, \underline {\hspace {1.25 ex}}\eta &= -\frac {x}{9}+y\right ] \\ \left [R &= -\frac {-27 y+9 x -2}{27 \left (3 x -1\right )^{\frac {3}{2}}}, S \left (R \right ) &= \frac {3 \ln \left (3 x -1\right )}{2}\right ] \\ \end{align*}