常数变易法在微分、差分方程中的应用

THE APPLICATION OF VARIATION OF CONSTANTS IN DIFFERENTIAL AND DIFFERENCE EQUATIONS

本文旨在研究常数变易法的思想根源,探索其在常微分方程和差分方程求解中的深入应用。常数变易法本质上是一种变量替换的思想.通过这种变量替换,可以将不容易直接利用初等积分法求解的复杂方程,转化成已知的、可求解的方程类型,进而求出原方程的通解.本文分别探讨了常数变易法在一阶非线性微分方程、高阶线性差分方程以及一阶向量差分方程求解中的应用,并给出了几类方程的求解方法和求解公式.

The purpose of this paper is to research the origin of variation of constant method and its deep application in solving nonlinear ordinary differential equations and linear difference equations.The substance of variation of constant method is a substitution of variable function.By this kind of substitution,we can transform the differential equation,which is unsolvable by elementary integration,to a new differential equation of the new variable functions which is solvable.This method is separately applied to first order nonlinear differential equations,higher order linear difference equations,and first-order vector difference equations.Some details of approaches and formulae of solution are given.