This paper presents the results of the numerical methods’ investigation for solving of ordinary differential equations and differential equations with delays. During this study were reviewed the following methods: explicit, implicit and symplectic Euler, Stormer-Verlet and trapezoid methods. Nowadays, there is a variety of different numerical methods, whose behavior for solving of ODE was well researched. In the first part there will be presented the analysis of permanence of the quadratic Hamiltonian of the ODE solutions by numerical methods, which were given above, in the second part - the similar researches, but for equations with delays. The obtained results for both of these types of systems will be compared.