Link to actual problem [4006] \[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right )=0} \]
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 (a -y \right ) \left (b -y \right )}}{\sqrt {f \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \arctan \left (\frac {\sqrt {f \left (x \right )}\, \left (y-\frac {a f \left (x \right )+f \left (x \right ) b}{2 f \left (x \right )}\right )}{\sqrt {-y^{2} f \left (x \right )+\left (a f \left (x \right )+f \left (x \right ) b \right ) y-f \left (x \right ) a b}}\right )\right ] \\ \end{align*}