Link to actual problem [8390] \[ \boxed {y^{\prime }-f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}=g^{\prime }\left (x \right ) f \left (x \right )} \]
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 {a g \left (x \right )+b}{g^{\prime }\left (x \right )}, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (g^{\prime }\left (x \right ) f \left (x \right ) a +g \left (x \right ) f^{\prime }\left (x \right ) a +f^{\prime }\left (x \right ) b \right )}{f \left (x \right ) g^{\prime }\left (x \right )}\right ] \\ \left [R &= \frac {y}{\left (a g \left (x \right )+b \right ) f \left (x \right )}, S \left (R \right ) &= \frac {\ln \left (a g \left (x \right )+b \right )}{a}\right ] \\ \end{align*}