This article investigates a novel method for time-varying systems with an impact term; we call it a modified homogenized highly precise direct integration method. Modified homogenized highly precise direct integration method can deal with the time-varying nonhomogeneous systems effectively by direct integration with high precision. Even though it is often difficult to select the time step size of integration properly for stiff problems, modified homogenized highly precise direct integration method can effectively deal with the this problem with a large time step size. By introducing new variants twice, modified homogenized highly precise direct integration method can easily deal with the nonhomogeneous term by a novel way, inherit the advantages of highly precise direct integration method and avoid the calculation of inverse matrix. The convergency and efficiency analyses are given in this article. Several numerical simulations and an application example are presented to demonstrate the high accuracy, eff...
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