A numerical method for solving the Cauchy problem for systems of ordinary differential equations by means of a system orthogonal in the sense of Sobolev generated by the cosine system

We consider iterative method that numerically solves Cauchy problem for systems of equations. Suggested method is based on using sobolev orthogonal system of functions, generated by cosine system $\{1, \sqrt{2}\cos(\pi k x), \; k \ge 1 \}$.