We consider the inverse problem for a first-order homogeneous system of linear ordinary differential equations (LODE),
where Y(t) is a vector-function with n components and A is an unknown matrix of dimensionality n × n with constant complex coefficients and certain restrictions imposed on its eigenvalues.
The boundary conditions are
Ck := Y(tk ), tk = t 0 + kd, d > 0, k = 0, 1, … , N, N ≥ n.
Here 
is a given system of vectors in 
.
This problem is equivalent to the problem of extrapolating a vector-function composed of quasi-polynomials representing solutions of some LODEs with constant coefficients of order n.
The zone of solution stability of the system against small-amplitude input data oscillations is described. The results obtained are used to construct an approximation algorithm for a real vector-function of one variable set at a finite number of nodes of a uniform grid (modified Prony algorithm).
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