2015 THREE(d)

The equation of motion of a particle is given by the differential equation

$\frac{d^{2}x}{d{t}^2}=-k^{2}x$

where $x$ is the displacement of the particle from the origin at time $t$, and $k$ is a positive
constant.

(i) Show that $x=A\cos kt+B\sin kt$, where $A$ and $B$ are constants, is a solution of the
equation of motion.

(ii) The particle was initially at the origin and moving with velocity $2k$.

Find the values of $A$ and $B$ in the solution $x=A\cos kt+B\sin kt$.