Solve the differential equation (dy)/(dx) + y tan x = sin 2x, when x = 0 and y = 1.

Linear equation $\frac{dy}{dx}+y\tan x=\sin 2x$ with $P=\tan x$. Integrating factor $=e^{\int\tan x\,dx}=e^{\log\sec x}=\sec x$. $y\sec x=\int 2\sin x\cos x\sec x\,dx=\int 2\sin x\,dx=-2\cos x+C$, so $y=-2\cos^2x+C\cos x$. Put $x=0,y=1$: $1=-2+C$, so $C=3$.