Link to actual problem [8464] \[ \boxed {x y^{\prime }+a y-f \left (x \right ) g \left (x^{a} y\right )=0} \]
type detected by program
{"unknown"}
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 {x^{1-a}}{f \left (x \right )}, \underline {\hspace {1.25 ex}}\eta &= -\frac {x^{-a} a y}{f \left (x \right )}\right ] \\ \\ \end{align*}