Link to actual problem [14394] \[ \boxed {-2 y^{\prime } t +2 y-\frac {1}{{y^{\prime }}^{2}}=-1} \]
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 &= \frac {3 t}{2}, \underline {\hspace {1.25 ex}}\eta &= y +\frac {1}{2}\right ] \\ \left [R &= \frac {2 y+1}{2 t^{{2}/{3}}}, S \left (R \right ) &= \frac {2 \ln \left (t \right )}{3}\right ] \\ \end{align*}