Geodesic
Properties of the integral curve and solving of non-autonomous system of ordinary differential equations
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2012), pp. 7-17

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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.
Keywords: system of ordinary differential equations, distribution probability-density function, integral curve, maximum movement.
Mots-clés : Liouville equation 
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G. A. Rudykh; D. J. Kiselevich. Properties of the integral curve and solving of non-autonomous system of ordinary differential equations. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2012), pp. 7-17. http://geodesic.mathdoc.fr/item/VSGTU_2012_2_a0/