On an approach to integration of ordinary differential equations with the use of series

A numerical analytic method of solving a Cauchy problem for linear and nonlinear 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 using Markov quadrature formulas with one or two fixed nodes. The method allows one to obtain an analytical 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 compared to the known Runge–Kutta, Adams, and Gear methods.