Problem

(a) Find the general solution of the differential equation

$$\frac{\mathrm{d}^2x}{\mathrm{d}t^2}+10\frac{\mathrm{d}x}{\mathrm{d}t}+25x=338\sin t.$$

(b) Show that, for large positive values of $t$ and for any initial conditions,

$$x\sim R\sin(t-\varphi),$$

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