Link to actual problem [4008] \[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) \left (y-c \right )=0} \]
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 {\sqrt {f \left (x \right ) \left (-y +c \right ) \left (b -y \right ) \left (a -y \right )}}{\sqrt {f \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {\sqrt {f \left (x \right )}}{\sqrt {f \left (x \right ) \left (-\textit {\_a} +c \right ) \left (-\textit {\_a} +b \right ) \left (a -\textit {\_a} \right )}}d \textit {\_a}\right ] \\ \end{align*}