Methods for numerical solutions of differential equations have been studied since the end of the last century. A large number of integration formulas have been published especially for solving special systems of differential equations. In general, it was not possible to choose the best method but for a subclass of tasks defined by similar properties the most suitable method could always be found. The presented "Modern Taylor Series Method" has proved to be both very accurate and fast. It is based on a direct use of the Taylor series.