We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.Consider the boundary-value problemwhere x.t; "/ is the required n-dimensional vector, t 2 OE0I 1; " 2 .0I " 0 / is a small parameter, h 2 N; A.t; "/ and B.t / are n n matrices, d is a p-dimensional column vector, f .t; "/ is an n-dimensional column vector, and M and N are p n matrices with constant elements. Assume that the following conditions are satisfied:(i) on the segment OE0I 1; the matrix A.t; "/ and vector f .t; "/ admit uniform asymptotic expansions in powers of the small parameter "; i.e.,(ii) the coefficients A k .t / and f k .t / of expansions (3) and the matrix B.t/ are infinitely differentiable on the segment OE0I 1I(iii) det B.t / D 0 8t 2 OE0I 1:Boundary-value problems of the type (1), (2) were investigated in [1][2][3][4][5] in the case where B.t/ D E; where E is the identity matrix. In [1,2], for the construction of asymptotic solutions of these problems, the method of regularization was used. In [3,4], the method of boundary functions was used for this purpose. In [5], the asymptotic behavior of a solution of a boundary-value problem was described using the idea of reduction of a linear system to an almost diagonal form. In [6], for the solution of this problem, the system was reduced to a system with a singular matrix coefficient of the derivative. In [7], a method was proposed for the asymptotic Nizhyn University, Nizhyn, Ukraine.

This paper deals with the boundary value problem for a singularly perturbed system of differential algebraic equations of the second order. The case of simple roots of the characteristic equation is studied. The sufficient conditions for existence and uniqueness of a solution of the boundary value problem for system of differential algebraic equations are found. Technique of constructing the asymptotic solutions is developed.

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It is investigated the possibility of construction of the asymptotic solution of the boundaryvalue problem for the linear singularly perturbed system of differential equations of the second order with identically degenerated matrix at the derivatives of higher order in case than boundary bundle of matrixes has simple spectrum. It was obtained the conditions of the existence and uniqueness of the solution of this boundary-value problem and its asymptotic is constructed in form of power series with degrees of small parameter.

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