An approximate analytical method of solving the systems of ordinary differential equations resolved with respect to the derivatives of unknown functions is considered. This method is based on the approximation of the solution to the Cauchy problem and its derivatives 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 formulas. This approach can be used to solve ordinary differential equations with a higher accuracy and with a larger discretization step compared to the known Runge-Kutta and Adams methods.

Keywords: ordinary differential equations, Cauchy problem, approximate analytical methods, numerical methods, orthogonal expansions, shifted Chebyshev series, Markov's quadrature formulas.

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