Link to actual problem [9168] \[ \boxed {y^{\prime }-\frac {\left (x^{4}+3 y^{2} x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(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 {x y}{6 y^{2}+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \ln \left (y\right )+\frac {3 y^{2}}{x}\right ] \\ \end{align*}