A boundary value problem for a linear integro-differential equation of order n+m with small parameters for m highest derivatives is considered under the condition that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to clarify the asymptotic behavior of solutions in the neighborhood of points where additional conditions are specified. The Cauchy function and boundary functions of the boundary value problem for a singularly perturbed homogeneous differential equation are constructed and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions of the boundary value problem is obtained. A theorem on an asymptotic estimate of the solution of the boundary value problem under consideration is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration has the phenomenon of an initial jump at the left end of this segment.