Numerical integration of ordinary differential equations using orthogonal expansions

We consider numerical 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 at each integration step into shifted Chebyshev series by Chebyshev polynomials of the first kind. Some relations connecting Chebyshev coefficients of the solution with Chebyshev coefficients of the right-hand side of the system being solved are obtained. Several equations for approximate values of Chebyshev 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 coefficients and for an approximate solution relative to the step size are given.