Link to actual problem [8886] \[ \boxed {{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right )=0} \]
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 &= f \left (x \right )^{-\frac {1}{n}} \left (f \left (x \right ) g \left (y \right )\right )^{\frac {1}{n}}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}f \left (x \right )^{\frac {1}{n}} \left (f \left (x \right ) g \left (\textit {\_a} \right )\right )^{-\frac {1}{n}}d \textit {\_a}\right ] \\ \end{align*}