  In the paper we offer a functional-discrete method for solving the Cauchy
problem for the first order ordinary differential equations (ODEs). This method
(FD-method) is in some sense similar to the Adomian Decomposition Method. But
it is shown that for some problems FD-method is convergent whereas ADM is
divergent. The results presented in the paper can be easily generalized on the
case of systems of ODEs.
