Vibration systems with time-varying mass are prevalent in engineering practice, exemplified by rockets with fuel depletion, vehicles with changing mass, and systems with cable-hoisted payloads. However, progress has been constrained by the lack of an end-to-end approach capable of integrating modeling, closed-form analysis, numerically stable calculations, and isolation design. Focusing on a typical system involving rocket fuel combustion with linear mass depletion, we first derive the equations of motion from the momentum theorem. A parameter transformation, constructed via the method of undetermined coefficients, converts the time-varying differential equation into a standard Bessel equation, yielding a closed-form analytical solution. To achieve reliable numerical solutions, a strategy combining variable upper-limit integration with grid-search-optimized lower bounds is deployed to replace indefinite integrals, thereby overcoming non-integrable products of Bessel functions. Benchmark comparisons with harmonic excitations show excellent agreement, validating the formulation and solution scheme. Building on the closed-form response, an end-to-end vibration isolation workflow is established via transmissibility and isolation failure-time-threshold (FTT) metrics, allowing for the selection of stiffness based on the instantaneous frequency ratio throughout the mass variation process. The resulting framework provides a general analytical tool for linear differential equations with time-varying mass and a practical pathway for vibration isolation design in mass-varying structures, with special relevance to aerospace applications.