Link to actual problem [15130] \[ \boxed {y-{y^{\prime }}^{2}+y^{\prime } x=x} \]
type detected by program
{"dAlembert"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _dAlembert]
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 {x}{2}-\frac {1}{2}, \underline {\hspace {1.25 ex}}\eta &= -\frac {x}{4}+y -\frac {1}{2}\right ] \\ \left [R &= -\frac {-4 y+1+2 x}{4 \left (-1+x \right )^{2}}, S \left (R \right ) &= 2 \ln \left (-1+x \right )\right ] \\ \end{align*}