Conditions guaranteeing asymptotic stability for the differential equation
$$\begin{aligned} x''+h(t)x'+\omega ^2x=0 \qquad (x\in \mathbb {R}) \end{aligned}$$
are studied, where the damping coefficient
\(h:0,\infty )\rightarrow 0,\infty )\) is a locally integrable function, and the frequency
\(\omega >0\) is constant. Our conditions need neither the requirement
\(h(t)\le \overline{h}<\infty \) (
\(t\in 0,\infty )\);
\(\overline{h}\) is constant) (“small damping”), nor
\(0< \underline{h}\le h(t)\) (
\(t\in 0,\infty )\);
\(\underline{h}\) is constant) (“large damping”); in other words, they can be applied to the general case
\(0\le h(t)<\infty \) (
\(t\in 0,\infty \))). We establish a condition which combines weak integral positivity with Smith’s growth condition
$$\begin{aligned} \int ^\infty _0 \exp -H(t)]\int _0^t \exp H(s)]\,\mathrm{{d}}s\,\mathrm{{d}}t=\infty \qquad \left( H(t):=\int _0^t h(\tau )\,\mathrm{{d}}\tau \right) , \end{aligned}$$
so it is able to control both the small and the large values of the damping coefficient simultaneously.