Link to actual problem [9244] \[ \boxed {y^{\prime }+\left (-\frac {\ln \left (y\right )}{x}+\frac {\cos \left (x \right ) \ln \left (y\right )}{\sin \left (x \right )}-f_{1} \left (x \right )\right ) y=0} \]
type detected by program
{"exactByInspection"}
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x y}{\sin \left (x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (y\right ) \sin \left (x \right )}{x}\right ] \\ \end{align*}