On the expressibility of the functions of the system x'(t) = A∙x(t), the eigenvalues of the matrix of which are non-multiple in the form of linear combinations of derivatives of one function included in this system

The problem of solving a system linear ordinary differential equations with constant coefficients is one the most important problems in both theory and algebra. Therefore, on hand, new methods algorithms are being developed for such systems, other existing systems improved. One well-known method reducing to single higher-order equation, which makes it possible find solutions original form combinations derivatives only unknown function.In this paper, we consider refinement general solution system; namely, study expressibility all functions homogeneous x^' (t)= A⋅x(t) function x_k (t), part system. For any matrix A, whose eigenvalues not multiples, simple criterion terms ranks formulated, its correctness proved. result obtained can also be applied periodicity complete observability.