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