Link to actual problem [4243] \[ \boxed {{y^{\prime }}^{3}-\left (a +b y+y^{2} c \right ) f \left (x \right )=0} \]
type detected by program
{"first_order_nonlinear_p_but_separable"}
type detected by Maple
[[_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 {{\left (\left (c \,y^{2}+b y +a \right ) f \left (x \right )\right )}^{\frac {1}{3}}}{f \left (x \right )^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {f \left (x \right )^{\frac {1}{3}}}{{\left (\left (\textit {\_a}^{2} c +\textit {\_a} b +a \right ) f \left (x \right )\right )}^{\frac {1}{3}}}d \textit {\_a}\right ] \\ \end{align*}