Abstract We study the problem of identifying the constant parameters involved in the right-hand sides of a linear non-autonomous system of differential equations with first-order ordinary derivatives. The specificity of the problem lies in the fact that additional conditions for identifying parameters, firstly, are non-local, and secondly, they include derivatives of an unknown function. We examine the conditions for the existence and uniqueness of a solution to the problem and propose two different approaches to the numerical solution of the problem. The results of computer experiments are presented.