### Abstract:

The article is devoted to the study of the asymptotic behavior of solving an integral boundary value problem
for a third-order linear differential equation with a small parameter for two higher derivatives, provided
that the roots of the "additional characteristic equation" have opposite signs. In the work are constructed
the fundamental system of solutions, boundary functions for singularly perturbed homogeneous differential
equation and are provided their asymptotic representations. An analytical formula of solution for a given
singularly perturbed integral boundary value problem is obtained. Theorem about asymptotic estimates of
solution is proved. For a singularly perturbed integral boundary value problem, the growth of the solution
and its derivatives at the boundary points of this segment is obtained when the small parameter tends to
zero. It is established that the solution of a singularly perturbed integral boundary value problem has initial
jumps at both ends of this segment. In this case, we say that there is a phenomenon of boundary jumps,
which is a feature of the considered singularly perturbed integral boundary value problem. Moreover, the
orders of initial jumps were different. Namely, at the point t = 0 , there is a phenomenon of the initial
jump of the first order, and at the point t = 1, the order of the initial jump was equal to zero. The results
obtained allow us to construct uniform asymptotic expansions of solutions of nonlinear singularly perturbed
integral boundary value problems.