This study addresses a class of inhomogeneous delay differential equations characterized by variable coefficients and the presence of two proportional delays. An exponential-type transformation is employed to convert the inhomogeneous formulation into an equivalent homogeneous pantograph equation. This reformulation enables the derivation of an explicit analytical solution represented by a superposition of an exponential function and a power series with coefficients given in closed product form. A detailed investigation of the convergence properties of the associated infinite product and series is conducted, yielding clear criteria for global convergence. Furthermore, specific parameter configurations are identified under which the series solution reduces to a finite sum, leading to exact analytical expressions. As an important application, the inhomogeneous Ambartsumian equation involving two proportional delays is examined, and new closed-form and exact solutions are established. Numerical experiments are included to support the theoretical analysis and to demonstrate the rapid convergence and the practical effectiveness of the proposed solution approach.

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.

Obtaining accurate solutions for mathematical models of neutron diffusion systems may lead to a deeper understanding of processes in reactor physics. The present paper applies the Laplace transform to the time-dependent neutron diffusion equation (together with the delayed neutron precursor equation) under a reflective boundary condition at one edge. The residue theorem is employed to obtain the inverse transform, leading to a series solution structured as a modal expansion associated with the eigenvalues of a transcendental equation. Moreover, the obtained series solution is theoretically proven to converge. The numerical results show acceptable accuracy based on residual errors. Physically, the neutron flux exhibits oscillatory behavior within the spatial domain, resulting in a wave-like alternating surface. Additionally, the delayed neutron precursor concentration stabilizes over time, gradually approaching a stationary profile, which is consistent with the physical expectations. The results also support the effectiveness of the Laplace transform technique in capturing the early-time behavior of the system. Differences between the present results and those reported in the relevant literature are explained.