Linear time-varying systems and bilinear systems are each analysed via Taylor series. Using the operational matrix for integration and the product operational matrix, the dynamical equation of a linear time-varying system (or a bilinear system) is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the Taylor series expansions can be determined recursively by the algorithm derived. The algorithm proposed here is similar to those already developed for orthogonal functions; however, owing to the simplicity of the operational matrix of integration and the product operational matrix, Taylor series present considerable computational advantages compared with the other polynomial series, provided that both the input and the output signals are analytic functions oft.