Link to actual problem [8391] \[ \boxed {y^{\prime }-a^{n} f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}=g^{\prime }\left (x \right ) f \left (x \right )} \]
type detected by program
{"unknown"}
type detected by Maple
[_Chini, [_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 {1}{g^{\prime }\left (x \right )}, \underline {\hspace {1.25 ex}}\eta &= \frac {y f^{\prime }\left (x \right )}{g^{\prime }\left (x \right ) f \left (x \right )}\right ] \\ \left [R &= \frac {y}{f \left (x \right )}, S \left (R \right ) &= g \left (x \right )\right ] \\ \end{align*}