Calculating the coefficients of an expansion of the solution to the Cauchy problem for ordinary differential equations in the form of a series in Chebyshev polynomials / O. B. Arushanyan, N. I. Volchenskova, S. F. Zaletkin. //Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2012. ? 5. P. 24-30 [Moscow Univ. Math. Bulletin. Vol. 67, N 5, 2012.].

Two approaches are proposed to determine an initial approximation for the coefficients of an expansion of the solution to a Cauchy problem for ordinary differential equations in the form of series in shifted Chebyshev polynomials of the first kind. This approximation is used in an analytical method to solve ordinary differential equations using orthogonal expansions.

Key words: ordinary differential equations, approximate analytical methods of integration, numerical methods of integration, orthogonal expansions, shifted Chebyshev polynomials, Mar\-kov quadrature formulas.