Average of Ordinary Differential Equations of the Second Order with Variable Factors

Abstract

Linear ordinary differential equations of the second order of a general view are considered with In variable factors (input equations) reduced to the self-interfaced form. The common decision Each input equation, by means of the integral formula, it is represented through the common decision The accompanying equation of the same type, but with constant factors. It is considered that the general solution of the accompanying equation is known. In the integral formula, except an accompanying solution, the fundamental solution of an input equation enters. From the integral formula implies two modes deriving of the approached analytical solution of an input equation. The first mode is based on Search of a fundamental solution by a method of perturbations. In the second mode assumed smoothness solutions of the accompanying equation and a possibility of its representation in a point of area in the form of a series Taylor through value of an accompanying solution in other point. As a result an initial solution it is represented in the form of a series on derivative of an accompanying solution. Factors at derivatives Are called as structural functions. They, actually, are the weighed moments fundamental solutions. Structural functions are continuous and are completely defined by functional dependence Initial factors from co-ordinates. They are converted in zero at constant initial factors, coinciding with accompanying factors. For determination of structural functions the system is received the recurrent equations. The well-founded mode of a choice of constants implies from this system Factors of the accompanying equation.