Hyers-Ulam Stability of Fifth Order Linear Differential Equations

Abstract

In this paper, we study the Hyers-Ulam stability for the fifth-order linear differential equation. In particular, we treat $\varsigma$ as an arrangement of differential equation and in the form
%\begin{equation*}
$$\varsigma^{v}(x)+\eta_1\varsigma^{iv}(x)+\eta_2\varsigma^{'''}(x)+\eta_3\varsigma^{''}(x)+\eta_4\varsigma^{'}(x)+\eta_5\varsigma(x)=\Omega(x)$$
%\end{equation*}
where $\varsigma \in c^{5} [k, l]$, $\Omega \in [k, l]$. We demonstrate that
$\varsigma^{v}(x)+\eta_1\varsigma^{iv}(x)+\eta_2\varsigma^{'''}(x)+\eta_3\varsigma^{''}(x)+\eta_4\varsigma^{'}(x)+\eta_5\varsigma(x)=\Omega(x)$ has the Hyers-Ulam stability. Two illustrative examples are given to represent the effectiveness of the proposed method. Fifth-order linear differential equations find applications in a wide range of fields, from engineering and control theory to physics, biology, and beyond. These equations are powerful tools for modeling systems with complex dynamics that involve multiple interacting forces or rates of change. Understanding and analyzing their stability and behavior can lead to significant advancements in the design, control, and optimization of these systems.