Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 2 / 12 / b / Solution

Set $\omega =\sqrt{k/m}$. Each coordinate satisfies the harmonic oscillator equation, so the general solution isThe four initial data give $A=X_0$, $B=C=0$, and $D=V_0/\omega$. ThusThis is the position component of the isotropic harmonic oscillator flow. Substitution also gives the constant energy $E=(k{X}_0^2+m{V}_0^2)/2$.