Use the method of characteristics to find $u(x, y)$ in the first quadrant $x \geqslant 0, y \geqslant 0$, where $u(x, y)$ satisfies

$$\frac{\partial u}{\partial x}-2 x \frac{\partial u}{\partial y}=\cos x$$

with boundary data $u(x, 0)=\cos x$.