Smooth solutions of linear functional differential equations of neutral type (Q1980782)

Consider a scalar functional differential equation of the form \[ \dot{y}(t)+p\dot{y}(t/q)=ay(t-1)+f(t),\quad t\in[0,+\infty) \] with initial condition \(y(t)=x^N(t)\), \(t\in[-1,0]\). We assume that \(q>1\), \(f(t)=\sum_{n=0}^F f_nt^n\) and \(x^N(t)=\sum_{n=0}^N x_nt^n\). Suppose that \(x^N(t)\) satisfies the equation \(\dot{x}(t)+p\dot{x}(t/q)=ax(t-1)+f(t)+f_Nt^N,\) \(t\in\mathbb{R}\). If \(p\ne-q^n\) for all integer \(n>0\), then the solution of the initial-value problem on the segment \([0,T]\), \(T>1\) has at the connection point of the solution continuous derivatives of degree not less than \(N\). Some numerical examples are given. For the entire collection see [Zbl 1467.34001].