= Solution
Set $\omega=\sqrt{k/m}$. Each coordinate satisfies the <harmonic oscillator equation>, so the <general solution> is
$$
x=A\cos\omega t+B\sin\omega t,\qquad
y=C\cos\omega t+D\sin\omega t.
$$
The four initial data give $A=X_0$, $B=C=0$, and $D=V_0/\omega$. Thus
$$
\boxed{x(t)=X_0\cos\omega t,\qquad y(t)=\frac{V_0}{\omega}\sin\omega t}.
$$
This is the position component of the <isotropic harmonic oscillator flow>. Substitution also gives the constant <energy> $E=(kX_0^2+mV_0^2)/2$.
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