The author considers the differential equation: \begin{equation} \tag{1} \ddot{q} + \omega^{2}(\epsilon \, t) \, q = f(t) \end{equation} where $\epsilon$ is a small number. He finds an approximate solution of (1) with an error $O(\epsilon^{2})$, when $f(t) = 0$, and with an error $O(\epsilon)$ when $f(t) \neq 0$. He exposes some applications of his results to a problem of mechanics.