On calculation of Chebyshev series coefficients for the solutions to ordinary differential equations

Canonical systems of second-order ordinary differential equations are considered. The solution of such a system and its derivatives are expanded in shifted series in Chebyshev polynomials of the first kind. Algorithms to determine initial approximations for the expansion coefficients are described. The resulting approximations are used in the method of approximate analytical integration of ordinary differential equations on the basis of Chebyshev series.