A new approach to solving differential equations based on a numerical experiment

The symmetry of the optimal algorithm for solving the HDE with respect to the point (x + h / 2) is shown by the numerical method. It was found that the error in recovering the antiderivative for periodic functions (Cos (x), Sin (x), etc.) is periodically zeroed in contrast to aperiodic functions (xp, Exp (x), etc.). Based on the symmetry of the arithmetic mean, formulas have been developed that make it possible to increase the accuracy of calculations by a factor of 100 in comparison with the fourth-order Runge-Kutta method.