Link to actual problem [3880] \[ \boxed {\left (x -6 y\right )^{2} y^{\prime }+2 y x -6 y^{2}=-a} \]
type detected by program
{"exact"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{\left (x -6 y \right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\left (x -6 y\right )^{3}}{18}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{2} y -6 x \,y^{2}+12 y^{3}+x a}{\left (x -6 y \right )^{2}}\right ] \\ \\ \end{align*}