Link to actual problem [4171] \[ \boxed {3 x^{4} {y^{\prime }}^{2}-y x -y=0} \]
type detected by program
{"first_order_nonlinear_p_but_separable"}
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 {\sqrt {3 x y +3 y}}{\sqrt {1+x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 y \sqrt {1+x}\, \sqrt {3}}{3 \sqrt {\left (1+x \right ) y}}\right ] \\ \end{align*}