General solutions of some linear equations with variable coefficients

Abstract

In this work we find general solutions to some classes of linear wave equations with variable coefficients. Such equations describe the oscillations of rods, acoustic waves, and also some models of gas dynamics are reduced to these equations. To construct general solutions, we employ special types of Euler-Darboux transformations, namely, Levi type transformations. These transformations are first order differential substitutions. For constructing each transformation, we need to solve two linear second order ordinary differential equations. The solutions of one of these equations are determined by the solutions of the other equations by means of a differential substitution and Liouville formula. In the general case, it is not easy to solve these ordinary differential equations. However, it is possible to provide some formula for the superposition of the transformation of Levy type. Starting with a classical wave equation with constant coefficients and employing the found transformations, we can construct infinite series of equations possessing explicit general solutions. By means of Matveev method we obtain limiting forms of iterated transformations. We provide a series of particular examples of the equations possessing general solutions.