The paper considers the linear ordinary differential system with constant coefficients (1) where A is a square matrix, p (t) are n-dimensional polynomial vectors with complex coefficients and complex numbers. Theorem 1 states an explicit solution of system (1) in finite terms. By means of this solution the Theorem 2 states a theoretical ground for the application of the method of indeterminate coefficients to the obtaining of a particular solution of system (1). Theorem 3 analyzes the very particular case in which k=l and is the only eigenvalue of A. Finally, a correspond¬ence is established between the differential systems (1) and the matrix equation
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