Summary: A semi-analytical time-integration procedure is presented for the integration of discretized dynamic mechanical systems. This method utilizes the advantages of boundary element method (BEM), well known from quasi-static field problems. Motivated by these spatial formulations, the present dynamic method is based on influence functions in time, and gives exact solutions in the linear time-invariant case. Similar to domain-type BEMs for nonlinear field problems, the method is extended for nonlinear and time-varying dynamic systems, where Duffing oscillator with time-varying mass is used as a representative model problem. The numerical stability and accuracy of the semi-analytical method are discussed in separated steps for time-varying masses and for nonlinear Duffing-type restoring forces. As an illustrative example, a Duffing oscillator with exponentially varying mass is studied in some detail. The case of a linear restoring force and an exponentially varying mass is compared to the closed form solution derived in the present paper. A sinusoidal variation of the mass in time is studied, too.