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.