Developed analytical approach for a special kind of differential-difference equation: Exact solution

This paper investigates a differential-difference equation with a variable coefficient of exponential order in the form φ′(t) = αφ(t) + βe^σt φ(−t). In literature, periodic solution has been obtained at the special case σ = 0. In this paper, an effective approach is developed to determine the exact solution in terms of exponential and trigonometric functions. In addition, the exact solution is expressed in terms of exponential and hyperbolic functions under specific conditions of the involved parameters. Exact solutions of several special cases are derived and found in full agreement with the corresponding results in the relevant literature. Some theoretical results are presented and proved which can be generalized to include other complex models. The behavior of the obtained shows periodicity in the absence of σ while the damped oscillations are shown graphically when σ is assigned to negative values.