Asymptotic properties of solutions to linear nonhomogeneous differential equations with variable delays and variable coefficients

In this paper, we examine the asymptotic properties of solutions to first-order linear nonhomogeneous differential equations with variable delays and variable coefficients. Initially, a unique global solution to the relevant characteristic equation is obtained. Subsequently, the asymptotic properties of the solutions to the delay equation are demonstrated through the utilization of the solution of the characteristic equation. Finally, upon examination of specific cases pertaining to the variable coefficients of the delay differential equation, three discrete results are obtained. The findings are expressed as the solution of the characteristic equation. In addition, seven examples are given to demonstrate the hypothesis of the main theorem.