= Solution
The <characteristic equation> is $r^2+\lambda r+k=0$, whose roots are $r_\pm=(-\lambda\pm\sqrt{\lambda^2-4k})/2$. For distinct real roots, $y=C_1e^{r_+t}+C_2e^{r_-t}$; at the repeated-root threshold $\lambda^2=4k$, $y=(C_1+C_2t)e^{-\lambda t/2}$. In the complex-root case put $\Omega=\sqrt{k-\lambda^2/4}$ to obtain
$$
\boxed{y=e^{-\lambda t/2}(C_1\cos\Omega t+C_2\sin\Omega t),\qquad\lambda^2<4k.}
$$
This is precisely the oscillatory regime of the <damped harmonic oscillator>. The zero solution is of course nonoscillatory; the regime describes its nontrivial motions.
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