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 $y\text{'}\text{'}+p\left(x\right)y^{\prime }+q\left(x\right)y=f\left(x\right)$, with condition that there exists a nonzero $y_1:I\to X$ in $C^2\left(I\right)$ such that ${y\text{'}\text{'}}_1+p\left(x\right)y_1^{\prime }+q\left(x\right)y_1=0$ 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.