韦达定理在推导常系数齐次线性微分方程通解式中的应用

Application of Vieta's theorem in deriving the general solution of homogeneous linear differential equations with constant coefficients

摘要

在探讨常系数齐次线性微分方程的通解时,通常侧重于通过求解代数方程即特征方程来找到方程的根,进而构建出通解.提出一种利用韦达定理来直接推导出这种微分方程通解形式的方法.通过利用韦达定理建立方程的系数与其根之间的直接联系,从而更为直观地构造出微分方程的通解.该方法为求解此类问题提供了一种新的思路.

Abstract

When discussing the general solution of homogeneous linear differential equations with constant coefficients, it is usually focused on finding the root of the equation by solving the algebraic equation, that is, the characteristic equation, to further construct the general solution. This paper proposed a method of directly deducing the general solution form of such differential equations by using Veda's theorem. By establishing a direct connection between the coefficients of the equation and its roots with Vieta's formulas, the general solution of the differential equation can be constructed more intuitively. This approach provides a new perspective for solving this type of problem.

关键词

韦达定理 / 常系数齐次线性微分方程 / 通解

Key words

Vieta's theorem / homogeneous linear differential equations with constant coefficients / general solution

参考文献

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