SOLUTION OF SOME LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

Article ID: 2559

Keywords: ordinary differential equations; linear systems; variable coefficients; integrable combinations; explicit form of the solution

Abstract

The problem of integrability of ordinary differential equations to find their exact solutions is a celebrated problem in the theory of differential equations which attracted attention of several workers in the area. This is due to the fact that: (a) differential equations are the most widely used continuous models of dynamic systems in physics, medicine, economics, biology and other sciences that study the surrounding reality, for which the explicit trajectory of the dynamic system’s behavior is important as the explicit solution contains in itself the maximum information about the behavior of the system; (b) an explicit solution of the equation is necessary to confirm the mathematical and physical intuition, to compare the solutions obtained by various approximate methods and to compare these methods. It is also worth noting that in the presence of various methods for obtaining an explicit form of solving differential equations, the advantage is given to simpler algorithms. This paper presents a method for finding an explicit form of the solution of one class of systems of linear ordinary differential equations of the first order with variable coefficients. Examples are given for illustration. This method includes elements of the well-known classical methods of the theory of integration of ordinary differential equations: the Leonard Euler method, based on the roots of the characteristic equation, and the Jean Leron D’Alembert method of integrable combinations.