This study deals with the Hyers--Ulam stability (HUS) of the second order linear differential equations $x''+\alpha x'+\beta x = f(t)$ without the assumption of continuity of $f(t)$. In particular, the main purpose of this study is to find a specific exact solution near the approximate solution, and the best HUS constant. Furthermore, the instability is also discussed, and a necessary and sufficient condition is obtained. Finally, a specific application example and a numerical simulation are presented.
