Summary: In this paper it is shown that under specified conditions on the initial data a certain infinite coupled system of ordinary differential equations has a solution satisfying an auxiliary convergence condition. The infinite system discussed is essentially the Galerkin expansion of the solution to a given quasi-linear e equation. The results obtained suffice to prove the existence of a solution to this equation of oscillations of a beam.

Keywords: quasi-linear wave equation, method of energy inequalities, equation of oscillations of a beam.

Summary. In this paper  we consider the Tricomi problem in a mixed domain for the Chaplygin equation. Frankl was first which  showed that the problem of the expiry of a supersonic jet from a vessel with plane walls (inside subsonic speed of the vessel) in the hodograph plane is reduced to the Tricomi problem for Chaplygin equation. In the method of auxiliary functions we received a new theorem of uniqueness of the solution of this problem with the Frankl condition in a new domain, without any restrictions, except  the smoothness on the elliptical part of the border domain.

Keywords: method of auxiliary functions, Chaplygin equation of the Tricomi problem, an equation of mixed type.