One method for the investigation of linear functional-differential equations

We consider a scalar linear retarded functional-differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \dot{x}(t)=ax\left( {t-1} \right)+bx\left( {\frac{t}{q}} \right)+f(t),\quad q>1. $$\end{document}The study of linear retarded functional-differential equations mainly deals with two types of initial-value problems: initial-value problem with initial functions and initial-value problems with initial point (when it is necessary to find a classical solution whose substitution in the original equation reduces it to the identity). In the paper, the initial-value problem with initial point is studied by the method of polynomial quasisolutions. We prove theorems on the existence of polynomial quasisolutions and exact polynomial solutions of the analyzed linear retarded functional-differential equation. The results of numerical experiments are presented.