A large number of differential equations can be reduced to polynomial form. As was shown in a number of works by various authors, one of the best methods for the numerical solution of the initial value problem for such ODE systems is the method of Taylor series. In this article we consider the Cauchy problem for the total linear PDE system, and then — a theorem about the accuracy of its solutions by this method is formulated and proved. In the final part of the article, four examples of total systems of partial differential equations to the well-known two-body problem are proposed: two of them are related to the Kepler equation, one to the motion of a point in the orbit plane, and the last to the motion of the orbit plane.