Analytical solution of a class of coupled second order differential-difference equations (Q1210469)

The authors study the coupled differential difference equation in the form (1) \(X'(t)+A_ 1X'(t)+A_ 2X'(t-w)+B_ 0X(t)=F(t)\), \(t\geq w\), \(X(t)=G(t)\), \(0\leq t\leq w\), where \(A_ 1,A_ 2,B_ 0\) are matrices in \(C^{n\times m}\), \(F(t)\in C^ n([w,\infty))\), \(G(t)\in C^ n([0,w])\). They construct the exact solution of the initial value problem (1) in an explicit way. The method proposed here is based on the concept of co- solution of the algebraic matrix equation (2) \(Z^ 2+A_ 1Z+B_ 0=0\). We say that \((X,T)\) is an \((n,q)\) co-solution of (2), if \(X\in C^{n\times q}\), \(X\neq 0\), \(T\in C^{q\times q}\) and \(XT^ 2+A_ 1XT+B_ 0X=0\).