Asymptotic estimates of the solution for a singularly perturbed Cauchy problem

The article focuses on the initial problem for a third-order linear integro-differential equation with a small parameter at the higher derivatives, assuming that the roots of the additional characteristic equation have opposite signs. This paper presents a fundamental set of solutions and initial functions for a singularly perturbed homogeneous differential equation. The solution to the singularly perturbed initial integrodifferential problem employs analytical formulas. A theorem concerning asymptotic estimates of the solution is established.