Asymptotics of the solution of one Valley–Ooussin problem with an unstable spectrum

A differential equation describes the relationship between an unknown function and its derivatives. Such connections are sought in various fields of knowledge: mechanics, physics, chemistry, biology, economics, sociology, oceanology, etc. Systems of ordinary differential equations with a small parameter are used in modeling processes of various natures. Typically, when modeling, small factors are discarded in order to obtain a simpler model from which the necessary information can be extracted. Practice has proven that small factors should be included not in equations, but in solutions. Equations containing small factors are called perturbed. Perturbation theory has been widely used in modern applied mathematics. With its help, researchers answer questions about the influence of various factors on the course of the process, about the stability of the obtained solutions, the proximity of the processes described by the obtained solutions to the real objects under study.
The article studies the Vall?e-Poussin problem for a system of inhomogeneous linear singularly perturbed ordinary differential equations of the first order. The peculiarity of the problem under consideration is that the spectrum of the matrix, which is the coefficient of the linear part of the system, is unstable at three points of the segment under consideration. It is required to construct a uniform asymptotic expansion of the solution to the problem, modifying the classical method of boundary functions.