Smooth solutions of some linear functional differential equations

The paper presents the results of studying the scalar linear functional differential equation of a delay type x(t) = a(t)x(t -1) + b(t)x(t / q) + f (t ), q > 1. The main attention is paid to the initial value problem with the initial function when the initial condition is specified at the initial set. The method of polynomial quasi-solutions, which is based on representation of the unknown function x(t) in the form of a polynomial of degree N, is applied as the research method. Substitution of this function into the original equation results in the residual A(t) = O(tN), for which a faithful analytical representation is obtained. In this case, the polynomial quasi-solution is understood as the exact solution in the form of the polynomial of degree N disturbed because of the residual of the original initial problem. It is proved that if for the initial value problem under study a polynomial quasi-solution of degree N is chosen as the initial function, then the solution generated will have smoothness of not less than degree N at abutment points.