Link to actual problem [8853] \[ \boxed {{y^{\prime }}^{3}-f \left (x \right ) \left (y^{2} a +b y+c \right )^{2}=0} \]
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 (f \left (x \right ) \left (a \,y^{2}+b y +c \right )^{2}\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 (f \left (x \right ) \left (\textit {\_a}^{2} a +\textit {\_a} b +c \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a}\right ] \\ \end{align*}