Problem

(a) Find the general solution of the differential equation

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

(b) For large positive values of $x$ and for any initial conditions, show that the solution to part (a) can be approximated by

$$y \approx R\sin(2x - \phi),$$

where $R$ and $\phi$ are positive constants to be determined.