Representation of Solutions of Linear Differential Systems of the Second Order with Constant Delays

Abstract

We deduce representations for the solutions of initial-value problems for n-dimensional differential equations of the second order with delays:

\( x^{{\prime\prime} }(t)=2 Ax^{\prime}\left( t-\tau \right)-\left({A}^2+{B}^2\right) x\left( t-2\tau \right) \)

and

\( x^{{\prime\prime} }(t)=\left( A+ B\right) x^{\prime}\left( t-\tau \right)- A B x\left( t-2\tau \right) \)

by using special delay matrix functions. Here, A and B are commuting (n × n)-matrices and τ > 0. Moreover, a formula connecting the delay matrix exponential function with delayed matrix sine and delayed matrix cosine is obtained. We also discuss common features of the considered equations.