One method for the investigation of linear functional-differential equations

Abstract

We consider the scalar linear retarded functional differential equation $$\dot{x}(t) = ax(t - 1)+ bx \left( \frac tq \right) + f(t), \quad q > 1.$$ The study of linear retarded functional differential equations deals mainly with two initial-value problems: an initial-value problem with initial function and an initial-value problem with initial point (when one seeks a classical solution whose substitution into the original equation reduces it to an identity). In the present paper, an initial-value problem with initial point is investigated by the method of polynomial quasisolutions. We prove theorems on the existence of polynomial quasisolutions and exact polynomial solutions of the considered linear retarded functional differential equation. The results of a numerical experiment are presented.