A new analytical approach for determining the exact solutions for free vibration of single-degree-offreedom (SDOF) systems with non-periodically time-varying coe$cients (mass and sti!ness) is presented herein. In this paper, the function for describing the variation of mass of a SDOF system with time is an arbitrary one, and the variation of the sti!ness is expressed as a functional relation with the mass function and vice versa. Using appropriate functional transform, the governing di!erential equation for the title problem is reduced to a Bessel's equation or other analytically solvable equations. Exact solutions for free vibration of SDOF systems with non-periodically varying coe$cients are obtained for six important cases. In order to simplify the free vibration analysis of a SDOF system with multi-step time-varying coe$cients, the fundamental solutions that satisfy the normalization conditions are constructed based on the exact solutions derived. It is more convenient to determine the displacement response of the SDOF system by using the fundamental solutions and a recurrence formula developed in this paper. Numerical example shows that the proposed procedure is a simple, e$cient and exact method.