In recent years, probabilistic-statistical methods for studying various problems of mechanics of solid deformable bodies are being applied more than ever. The principal part of its field of application is the development of a general theory of strength and a hereditarily deformed solid. As known, the theory of random oscillations is increasingly being used in technology. Current research is aimed to present a numerical-analytical approach for studying the dynamic response of a hereditarily deformable system to unsteady input influences. It is established that the dynamic reaction of hereditarily deformable systems to an arbitrary form of random perturbations can also be represented as the Duhamel integral if the impulse transition function satisfies special Cauchy problems for the integro-differential equation (IMU). A study proposes an accurate analytical solution to the IMU of an impulsive transition function in existing weakly singular Rzhanitsyn-Koltunov nucleuses.