Link to actual problem [4312] \[ \boxed {{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{4} \left (y-b \right )^{3}=0} \]
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 \right )^{4} \left (b -y \right )^{3}\right )^{\frac {1}{6}}}{f \left (x \right )^{\frac {1}{6}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {f \left (x \right )^{\frac {1}{6}}}{\left (f \left (x \right ) \left (a -\textit {\_a} \right )^{4} \left (-\textit {\_a} +b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a}\right ] \\ \end{align*}