ANALYSIS OF THE GENERAL SOLUTION OF ONE LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION WITH REGULAR SINGULAR POINT

Abstract

The processes are studied in the applied fields can be described by differential equations, the structure of which is different from the structure of classical differential equations, which have currently well understood the methods of solving. Finding the Kummer-Liouville transform for a certain pair of (even linear) differential equations is related to the solution of the non-linear differential equation Yermakov, which has an analytical solution in a limited number of cases of its coefficients. Therefore, the reduction of a specific ordinary differential equation (which has practical interest in a particular field of application) to a known type of differential equation remains an urgent task for research. It is established that the linear homogeneous second-order differential equation that has the regular singular point chosen for the study is reduced to a confluent hypergeometric equation presented in the form of Whittaker using the Kummer-Liouville transform in the work. Relations between the coefficients of the given differential equation and the confluent hypergeometric equation in the general form and in the Whittaker form are found. Based on these relations, it is shown that the equation under study has only one independent fundamental solution in the form of a confluent hypergeometric function. The second fundamental solution can be found either in the form of a generalized power series using the Frobenius method, or can be expressed through the Tricomi function. In the latter case, the general solution of the equation under study is a linear combination of Whittaker functions. Each fundamental solution is a series of special kind which is centered at a regular singular point of the differential equation. The possibility of the adoption by the Tricomi function of a finite expression in the form of a polynomial with respect to the variable x or 1/x for physically plausible values of the coefficients of the differential equation under study is inspected.