Link to actual problem [12007] \[ \boxed {{\mathrm e}^{-y} \sec \left (x \right )-{\mathrm e}^{-y} y^{\prime }=-2 \cos \left (x \right )} \]
type detected by program
{"exactWithIntegrationFactor"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(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}{2 \left (\sec \left (x \right )+\tan \left (x \right )\right ) \cos \left (x \right )}, \underline {\hspace {1.25 ex}}\eta &= \frac {\sec \left (x \right )}{2 \left (\sec \left (x \right )+\tan \left (x \right )\right ) \cos \left (x \right )}\right ] \\ \\ \end{align*}