Polynomial quasisolutions of linear differential-difference equations

Abstract

The paper discusses a linear differential-difference equation of neutral type with linear coefficients, when at the initial time moment $t=0$ the value of the desired function $x(t)$ is known. The authors are not familiar with any results which would state the solvability conditions for the given problem in the class of analytical functions. A polynomial of some degree $N$ is introduced into the investigation. Then the term »polynomial quasisolution« (PQ-solution) is understood in the sense of appearance of the residual $\Delta (t)=O(t^N)$, when this polynomial is substituted into the initial problem. The paper is devoted to finding PQ-solutions for the initial-value problem under analysis.