Reliable analytical approach to solving delay differential equations with combined proportional and constant delays

The investigation of delay differential equations (DDEs) spans a diverse array of practical applications. In the realm of applied sciences, DDEs typically involve either constant/pure delays or proportional delays, each posing significant analytical challenges. The task of deriving exact solutions becomes increasingly intricate when both types of delays are integrated within a single model. This paper derives an explicit unified analytical solution for a DDE with combined delays using the method of steps (MoS). A unified solution formula is presented, applicable across any sub-interval of the problem's domain. Additionally, the theoretical properties of the solution and its derivative—such as continuity and the presence of discontinuities at specific points—are meticulously examined. The proposed methodology also encompasses existing findings in the literature, with applications extending to fields including astronomy and railway electrification.