Ulam–Hyers stability of some linear differential equations of second order

Abstract

In this work we prove the Ulam–Hyers stability of the following equation $\left(E\right){\varphi }^{\prime \prime }\left(x\right)+(\gamma -1){\varphi }^{\prime }\left(x\right)-\gamma \varphi \left(x\right)=0,$where $\gamma$ is a real number. The main purpose is to find a solution $\varphi$ of (E) satisfying $|\varphi \left(x\right)-f\left(x\right)|⩽K\varepsilon$, where $K$ is Ulam–Hyers-Stability constant and $f$ is an exact solution of the associated inequality $|f^{\prime \prime }\left(x\right)+(\gamma -1)f^{\prime }\left(x\right)-\gamma f\left(x\right)|⩽\varepsilon ,$for any $\varepsilon >0$.