Problem

(a) Find the general solution of the differential equation

$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 4\frac{\mathrm{d}y}{\mathrm{d}x} + 3y = 5\cos x.$$

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

$$y \approx R\sin(x + \phi),$$

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