Abstract
The main difficulty of the Cauchy equation is that there is no domain in which the desired solution must be determined.This situation leads to the complexity of finding the answer.The study presents a solution of the Cauchy problem at any values.The achievement of the set goal will enable solving one of the key problems of gas dynamics.To that end, it is necessary to define the solution, since a solution in a classical form is nonexistent for this type of equations.We presented an algorithm for the construction of periodic, in terms of variables, solutions of the system in first order partial derivatives.The study found a sufficient condition of existence of periodic, in terms of variables, solutions in the broad sense of differential equation systems in partial derivatives.