Abstract: In this paper, we get a necessary and sufficient condition such that a class of differential inequalities hold. Using this necessary and sufficient condition, we prove that a class of first order nonhomogeneous ordinary differential equations have the Hyers-Ulam stability. And then, we prove that some first order nonhomogeneous ordinary differential equations and some second order nonhomogeneous ordinary differential equations do not have the Hyers-Ulam instability under some suitable conditions.

Key words: Hyers-Ulam stability, Hyers-Ulam instability, Ordinary differential equations; Necessary and sufficient condition