Let y=y(x) be the solution of the differential

$\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x$

$\frac{d y}{d x}+\frac{2 \sin x}{\cos x} y=2 \sin x$

I.F. $=e^{\int 2 \frac{\sin x}{\cos x} d x}$

$=e^{2} \ln \sec x=\sec ^{2} x$

$y \cdot \sec ^{2} x=\int 2 \sin x \cdot \sec ^{2} x d x$

$y \sec ^{2} x=2 \int \tan x \sec x d x$

$y \sec ^{2} x=2 \sec x+c$

At $x=\frac{\pi}{3}, y=0$

$\Rightarrow 0=2 \sec \frac{\pi}{3}+C \Rightarrow C=-4$

$y \sec ^{2} x=2 \sec x-4$

Put $x=\frac{\pi}{4}$

$y \cdot 2=2 \sqrt{2}-4$

$y=\sqrt{2}-2$