Two basic iterative solving methods of Cauchy problem of the first order equations

In this paper by employing similar standard methods, the theorem of two essential iterative processes namely, Pickard and Newton´s applicable to Cauchy´s problem for the first order ordinary differential equations have been proved. Those methods permit to compare the mentioned processes by both its convergence acceleration and by its segment length convergence. It has been demonstrated that, the iteration calculated by Newton´s method, incomparably excessive rapidity approach to the exact solution. In the same time the segment lengths for which the given iterative process converges, do not diverge too much from each other. The application of the solution method of the general Ricatti´s equation with acquired numerical results, developed by the authors has been revealed.