METHOD OF SOLVING LINEAR DIFFERENTIAL EQUATIONS OF THE THIRD AND FOURTH ORDERS WITH VARIABLE COEFFICIENTS

Abstract

The mathematical model makes it possible to study the phenomenon as a whole, to predict its development, to make quantitative estimates of changes occurring in the system over time. There is a large number of analytical and numerical me-thods for solving ordinary differential equations and their systems, by which one can simulate the dynamics of oscillatory processes. The number of physical problems that are solved with the help of differential equations of the third or fourth order is constantly growing. Such problems arise where there are uneven transitions from one physical characteristic to another, including questions of quantum mechanics, astrophysics, and the theory of elasticity. Unfortunately, for many practically important questions, the problems described by differential equations are very complex, and their exact solution is difficult or impossible. Since these solutions describe those or other natural processes, for the researcher important information is about them. Therefore, getting the exact solution in an analytical form is an important and actual problem. A new method for solving linear homogeneous differential equations of the third and fourth orders with variable coefficients was constructed, which allowed us to obtain an analytic solution throughout the field of equation determination. The proposed method reduces the order of the differential equation and is a generalization of Abel's formula for the case of linear differential equations of the third and fourth orders with variable coefficients. The developed method is based on the use of the properties of determinant Vronsky. The article gives the conditions for which determinant can be expressed through elementary functions. The conditions that ensure the possibility of using the proposed method are explored and substantiated. A recurrence formula has been found to obtain a partial solution that simplifies the process of solving these equations.