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This paper proposes a method of obtaining a numerical, exact discrete-time-model for linear time-varying systems. The method relies on the computation of a transition matrix expressible as the Peano-Baker series for a given discrete-time interval and system parameters. For time-invariant systems, the proposed discrete-time model reduces to the well known step-invariant-model. As an example, the Euler differential equation is discretized using the standard forward-difference method, the discretization of Euler differential operator, and the proposed method. Simulations show that the proposed discrete-time-model gives exact values at discrete-time instants for any discretization periods, while the other two methods generate errors.