About one Approach to a Solution of Linear Differential Equations with Variable Coefficients

Abstract

Using the uniform approach, linear differential equations of elliptic, hyperbolic and parabolic types with variable coefficients depending on coordinates and time are considered. It is shown that the solution of the initial equation can be expressed by means of an integral formula through the solution of the accompanying equation of the same type, but with constant coefficients. It is considered that the solution of the accompanying equation is known. From the integral formula, assuming smoothness of the accompanying solution, an equivalent representation of the solution of the initial equation is obtained in the form of series in all possible derivatives of the solution of the accompanying equation. For coefficients at derivatives a system of recurrent equations is obtained, which can be solved analytically at some cases.