# Abstract

In this paper, we consider non-autonomous system of ordinary diﬀerential equations. For a given non-autonomous system, we introduce the distribution probability-density function of representative points of the ensemble of Gibbs, possessing all the characteristic properties of the probability-density function, and satisfying the partial differential equation of the ﬁrst order (Liouville equation). It is shown that such distribution probability-density function exists and represents the only solution of the Cauchy problem for the Liouville equation. We consider the properties of the integral curve and the solutions of non-autonomous system of ordinary diﬀerential equations. It is shown that under certain assumptions, the motion along trajectories of the system is the maximum of the distribution probability-density function, that is, if all the required terms are satisﬁed, an integral curve of non-autonomous system of ordinary differential equations at any given time is the most probable trajectory. For the linear non-autonomous system of ordinary differential equations, it is shown that the motion along the trajectories is carried out in the mode of distribution probability-density function and the estimate of its solutions is found.

# About the authors

### Gennady A Rudykh

Institute of Mathematics, Economics and Computer Science of Irkutsk State University

Email: rudykh@icc.ru
(д.ф.-м.н., проф.), профессор, каф. математического анализа и дифференциальных уравнений; Институт математики, экономики и информатики Иркутского государственного университета; Institute of Mathematics, Economics and Computer Science of Irkutsk State University

### Daria J Kiselevich

Institute of Mathematics, Economics and Computer Science of Irkutsk State University

Email: dariakis@mail.ru
аспирант, каф. математического анализа и дифференциальных уравнений; Институт математики, экономики и информатики Иркутского государственного университета; Institute of Mathematics, Economics and Computer Science of Irkutsk State University

# References

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