Ыmooth solutions to some differential-difference equations

In this paper, we consider a scalar linear differential-difference equation (LDDE) of neutral type $\dot{x}(t)+p(t)\dot{x}(t-1)=a(t)x(t-1)+f(t)$. We examine the initial-value problem with an initial function in the case where the initial condition is given on an initial set. We use the method of polynomial quasisolutions based on the representation of the unknown function $x(t)$ in the form of a polynomial of degree $N$. Substituting this function in the original equation we obtain the discrepancy $\Delta(t)=O(t^{N})$, for which an exact analytic representation is obtained. We prove that if a polynomial quasisolution of degree $N$ is taken as an initial function, then the smoothness of the solution generated by this initial functions at connection points in no less than $N$.