= Solution
Assume a physical <mass> $m>0$, <spring constant> $k>0$, and nonnegative viscous damping parameter $\gamma$. The homogeneous <damped harmonic oscillator> has <characteristic equation>
$$
r^2+\gamma r+\omega_0^2=0,\qquad\omega_0^2=\frac{k}{m}.
$$
Its roots are $r=-\gamma/2\pm\sqrt{\gamma^2/4-\omega_0^2}$. They have nonzero imaginary parts precisely when $\gamma^2<4k/m$. Defining $\Omega=\sqrt{k/m-\gamma^2/4}$, the real oscillatory solutions are
$$
\boxed{z_h(t)=e^{-\gamma t/2}\big(C\cos\Omega t+D\sin\Omega t\big),\qquad\gamma^2<\frac{4k}{m}.}
$$
For strictly decaying oscillations also require $\gamma>0$. The limiting $\gamma=0$ case is an undamped <harmonic oscillator>. Equality gives critical damping with $z_h=(C+Dt)e^{-\gamma t/2}$, and a larger positive $\gamma$ gives two real decaying exponentials rather than oscillations.
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