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

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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.

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Authors and Affiliations

  1. Brno University of Technology Brno, Brno-střed, Czech Republic

    Z. Svoboda

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  1. Z. Svoboda
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Correspondence to Z. Svoboda.

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Published in Neliniini Kolyvannya, Vol. 19, No. 1, pp. 129–141, January–March, 2016.

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Svoboda, Z. Representation of Solutions of Linear Differential Systems of the Second Order with Constant Delays. J Math Sci 222, 345–358 (2017). https://doi.org/10.1007/s10958-017-3304-9

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  • DOI: https://doi.org/10.1007/s10958-017-3304-9

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