A unified analytical treatment for solving delay differential equations: exact solution

The delay differential equations (DDEs) are widely used to explore various engineering and physical applications. An example of DDEs with proportional delays is known as the pantograph model which governs the current collection in electric trains. DDEs with constant delays also have different applications. This paper introduces a unified approach to analyze a class of first order DDEs under arbitrary history functions (HFs). The proposed approach assumes that the arbitrary HF \(\phi(t)\) can be represented as Maclaurin series with coefficients \(\phi_m, ~m\ge0\). Based on this assumption, the solution in each sub-interval of the problem's domain is obtained in explicit form in terms of the coefficients \(\phi_m\). Exact solutions are obtained for several examples subjected to history functions of different forms. Properties of the solution and its derivative are proved and examined theoretically. Existing results in the literature are derived from the current ones as special cases. In view of the obtained results, the exact solution of any first order linear delay differential equation can be directly determined once the coefficients \(\phi_m\) of the given history function is inserted into the standard solution. This reflects the advantage of the proposed approach over other techniques. Moreover, the suggested analysis can be easily extended to include higher order linear delay models.