Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations

Abstract

We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential equation of the form 𝑦+𝑝(𝑥)𝑦+𝑞(𝑥)𝑦=𝑓(𝑥), with condition that there exists a nonzero 𝑦1∶𝐼→𝑋 in 𝐶2(𝐼) such that 𝑦1+𝑝(𝑥)𝑦1+𝑞(𝑥)𝑦1=0 and 𝐼 is an open interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam stability of several important well-known differential equations.