Insight on Generalizing Forced Proportional Delay Equations via a Developed Analytical Solution

This paper develops a unified analytical approach for a broad class of delay differential equations with proportional delay and an arbitrary forcing term. The models under consideration incorporate a proportional delay characterized by a contraction of the independent time variable, a structure that arises naturally in various applied settings. The proposed approach is based on power series expansions, which yield explicit recursive relations for the solution coefficients and provide a systematic procedure for constructing the corresponding solutions. The resulting solution is naturally decomposed into two complementary components: one associated with the underlying homogeneous delay equation and the other representing the contribution of the external forcing function. In addition, rigorous convergence analysis is presented, establishing the absolute convergence of the resulting series representations. Building upon the general formulation, several exact closed-form solutions are derived for important classes of forcing functions, including polynomial, periodic , and exponential functions. A range of distinguished and limiting cases is also investigated to illustrate the versatility, consistency, and unifying character of the proposed methodology. The results extend and encompass several existing findings in the literature while providing a systematic and effective analytical tool for a broad class of forced delay models.