Explanation

(a)
Here, $x = A\cos{4t + B\sin{4t\ }}$
On differentiating $w.r.t.t,$ we get
$\frac{dx}{dt} = - 4A\sin{4t + 4B\cos{4t}}$
Again, on differentiating $w.r.t.t,$ we get
$\frac{d^{2}x}{dt^{2}} = - 16A\cos{4t - 16B\sin{4t}}$
$= - 16(A\cos{4t + B\sin{4t)}}$
$\Rightarrow \ \frac{d^{2}x}{dt^{2}} = - 16x$