A vector space approach for generating the response of linear multi-degree-of-freedom time-invariant and time-varying dynamic systems using Hamilton's law of varying action (HLVA) is presented. The boundary (initial) condition constraints on the temporal-basis-function expansions of the time-dependent variables has been removed, while preserving HLVA in its original form. This provides for the broadest choice of basis functions. As a result of this new approach, it has been demonstrated that the response of dynamic systems are composed of temporal modes herein denoted as fundamental-time modes (FTM). Using these fundamental-time-modes, the general solution for the system response is obtained without reference to initial conditions or forcing functions. The unique response of the system is subsequently generated by using the initial conditions and forcing functions to scale the FTM. This new approach is demonstrated to provide the exact analytical response, as well as provide accurate numerical response solutions to a forced-damped-spring}mass system, using admissible temporal-basis functions (TBF). The numerical response solutions using Gaussian radial basis functions, are compared to those obtained by using power-series and third order Hermite polynomials. The new methodology, which will be referred to as the universal method, in conjunction with Gaussian TBF has permitted use of transition intervals (ordinarily referred to as time steps) of unprecedented length (larger than the period of the motion) while still maintaining an accurate response solution.