Link to actual problem [4009] \[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )=0} \]
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 &= \frac {1}{\sqrt {f \left (x \right )}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \int \sqrt {f \left (x \right )}d x\right ] \\ \end{align*}