DOI: https://doi.org/10.36719/2663-4619/119/74-82
Saltanat Veysova
National Defense University
PhD in Physics and Mathematics
https://orcid.org/0009-0003-9379-3280
seltenet.veysova63@gmail.com
Sima Pashayeva
National Defense University
PhD in Physics and Mathematics
https://orcid.org/0009-0008-0079-1726
sima.pashayeva73@gmail.com
Uniqueness Criteria for the Solution of Autonomous Differential Equations
Abstract
The article considers the uniqueness of solutions of autonomous differential equations. The uniqueness and continuability of solutions Initial Value Problem (IVP) for functions whose zeros divide the number line into intervals where the sign of the function is constant and are determined by the choice of the initial point on each interval are investigated. In studying the existence of a special decision to this type of differential equation, “one-sided” uniqueness was established depending on the given function and the location of its zeros in the coordinate plane. It is shown that the time it takes for the solution of the equation to pass from the neighborhood of the initial point to this point is determined by an improper integral. It has been studied that the violation of the uniqueness of the solution at the final moment of time is due to the fact that the improper integral converges and the impossibility of the solution coming to the equilibrium position is due to the fact that the improper integral diverges, there by preserving the uniqueness.
Keywords: Autonomous differential equation, the existence and uniqueness of the solution, Lipschitz condition, particular solution, "one-sided" uniqueness, direction fields, integral curve, phase portrait