Numerical integration of ordinary differential equations with the use of Chebyshev's series

We consider numerical analytical methods of approximate solving Cauchy problems for systems of ordinary differential equations of first and second orders. These methods are based on the expansions of the solution and its derivative into shifted Chebyshev's series at each integration step by Chebyshev's polynomial of the first kind. Some relations connecting Chebyshev's coefficients of the solution with Chebyshev's coefficients of the right-hand side of the system being solved are obtained. A representation of the solution as a functional series derived on the basis of integrals of Chebyshev's polynomials is studied. A number of equations for approximate values of Chebyshev's coefficients for the right-hand side of the system are deduced. An iterative process of their solution is described. Some error estimates for approximate Chebyshev's coefficients and for an approximate solution relative to the step length are given.