[tex]\dfrac{\mathrm dy}{\mathrm dx}+y\cos x=5\cos x[/tex]
[tex]e^{\sin x}\dfrac{\mathrm dy}{\mathrm dx}+ye^{\sin x}\cos x}=5e^{\sin x}\cos x[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[e^{\sin x}y\right]=5e^{\sin x}\cos x[/tex]
[tex]e^{\sin x}y=5\displaystyle\int e^{\sin x}\cos x\,\mathrm dx[/tex]
[tex]e^{\sin x}y=5e^{\sin x}+C[/tex]
[tex]y=5+Ce^{-\sin x}[/tex]
With [tex]y(0)=7[/tex], we have
[tex]7=5+Ce^{-\sin 0}\implies 7=5+C\implies C=2[/tex]
so that the particular solution is
[tex]y=5+2e^{-\sin x}[/tex]
