\[ y'(x)+y(x) \tan (x)-\sin (2 x)=0 \] ✓ Mathematica : cpu = 0.029944 (sec), leaf count = 17
\[\left \{\left \{y(x)\to -2 \cos ^2(x)+c_1 \cos (x)\right \}\right \}\] ✓ Maple : cpu = 0.023 (sec), leaf count = 13
\[y \relax (x ) = \left (-2 \cos \relax (x )+c_{1}\right ) \cos \relax (x )\]
Hand solution
\begin {equation} \frac {dy}{dx}+y\relax (x) \tan \relax (x) =\sin \left (2x\right ) \tag {1} \end {equation}
Integrating factor \(\mu =e^{\int \tan dx}=e^{-\ln \left (\cos \relax (x) \right ) }=\frac {1}{\cos \relax (x) }\). Hence (1) becomes
\[ \frac {d}{dx}\left (y\relax (x) \frac {1}{\cos \relax (x) }\right ) =\frac {1}{\cos \relax (x) }\sin \left (2x\right ) \]
Integrating both sides
\begin {align*} y\relax (x) \frac {1}{\cos \relax (x) } & =\int \frac {1}{\cos \relax (x) }\sin \left (2x\right ) dx+C\\ y\relax (x) & =\cos \relax (x) \int \frac {\sin \left (2x\right ) }{\cos \relax (x) }dx+C\cos \relax (x) \end {align*}
But \(\sin \left (2x\right ) =2\sin \relax (x) \cos \relax (x) \) hence
\begin {align*} y\relax (x) & =\cos \relax (x) \int \frac {2\sin \relax (x) \cos \relax (x) }{\cos \relax (x) }dx+C\cos \relax (x) \\ & =2\cos \relax (x) \int \sin \relax (x) dx+C\cos \relax (x) \\ & =-2\cos ^{2}\relax (x) +C\cos \relax (x) \end {align*}