Polynomial quasisolutions of linear second-order differential-difference equations

Abstract

The second-order scalar linear difference-differential equation (LDDE) with delay $$\ddot{x}(t) + (p_0+p_1t)\dot{x}(t) = (a_0 +a_1t)x(t-1)+f(t)$$ is considered. This equation is investigated with the use of the method of polynomial quasisolutions based on the presentation of an unknown function in the form of polynomial $x(t)=\sum_{n=0}^{N}x_n t^n.$ After the substitution of this function into the initial equation, the residual $\Delta(t)=O(t^{N-1}),$ appears. The exact analytic representation of this residual is obtained. The close connection is demonstrated between the LDDE with varying coefficients and the model LDDE with constant coefficients whose solution structure is determined by roots of a characteristic quasipolynomial.