Verification of approximate solutions for nonlinear differential equations

Equations for checking the approximate solutions of nonlinear differential equations are given. These solutions should predict the behavior of the systems for which these differential equations are the mathematical models. It is general consensus, regardless of which integration method is used to determine approximate functions, it must be checked whether these functions satisfy control equations with the desired accuracy in order to be considered approximate solutions. Here, the control equations are the differential equations of motion and the balance equation for the variation of the kinetic energy and the mechanical work of the forces. The errors in these control equations, residues respectively deviations, must be less than desired limits. As example, the sinking motion of a mass point in a liquid medium with the drag force assumed to be proportional to the square of the velocity is considered. Therefore, the differential equation of motion is nonlinear. The exact solution is determined and compared with the approximate function established with Euler’s polygon method. It turns out that for integration steps larger than steps for which bifurcation occurs the residues and the deviations are too large and such functions are not approximate solutions and cannot be used for making predictions about the behavior of the system. The energetic interpretation of the result is given.