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In the paper a new numerical-analytical method for solving the Cauchy problemfor systems of ordinary differential equations of special form is presented.The method is based on the idea of the FD-method for solving the operatorequations of general form, which was proposed by V.L. Makarov. The sufficientconditions for the method converges with a superexponential convergence ratewere obtained. We have generalized the known statement about the localproperties of Adomian polynomials for scalar functions on the operator case.Using the numerical examples we make the comparison between the proposed methodand the Adomian Decomposition Method.

A Superexponentially Convergent Functional-Discrete Method for Solving the Cauchy Problem for Systems of Ordinary Differential Equations