Link to actual problem [3390] \[ \boxed {y^{\prime }-x^{m -1} y^{-n +1} f \left (a \,x^{m}+b y^{n}\right )=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
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 &= x^{-m +1}, \underline {\hspace {1.25 ex}}\eta &= -\frac {a \,y^{-n +1} m}{n b}\right ] \\ \left [R &= \frac {a \,x^{m}+b y^{n}}{b}, S \left (R \right ) &= \frac {x^{m}}{m}\right ] \\ \end{align*}