A numerical analytic method of solving the Cauchy problem for normal systems of ordinary differential equations is proposed. The method is based on the approximation of the solution and its derivative by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process with the use of Markov's quadrature with one or two fixed nodes. The method yields an analytic representation of the solution and its derivative and can be used to solve ordinary differential equations with a higher accuracy and with a larger discretization step in comparison to the Runge-Kutta and Adams methods.

Keywords: ordinary differential equations, approximate analytic methods, numerical methods, orthogonal expansions, shifted Chebyshev series, Markov's quadrature formulas

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