Hyers-Ulam stability of first-order homogeneous linear differential equations with a real-valued coefficient

This paper is concerned with the Hyers–Ulam stability of the first-order linear differential equation x′ − ax = 0, where a is a non-zero real number. The main purpose is to find an explicit solution x(t) of x′−ax = 0 satisfying |φ(t)−x(t)| ≤ ε/|a| for all t ∈ R under the assumption that a differentiable function φ(t) satisfies |φ′(t)− aφ(t)| ≤ ε for all t ∈ R. In addition, the precise behavior of the solutions of x′ − ax = 0 near the function φ(t) is clarified on the semi-infinite interval. Finally, some applications to nonhomogeneous linear differential equations are included to illustrate the main result. © 2016 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).