谈常系数齐次线性微分方程组的标准基解矩阵

On the Standard Fundamental Matrix for Homogeneous Linear Differential Systems with Constant Coefficients

摘要: 常系数齐次线性微分方程组是微分方程理论的重要组成部分，其标准基解矩阵能够反映系统的动态行为和稳定性。因此，标准基解矩阵的求解是从理论分析迈向工程应用的关键步骤。本文以常系数齐次线性微分方程组为研究对象，旨在给出其全体标准基解矩阵的统一表达式，并将该表达式应用于简谐振子的自由振动问题。

Abstract: Systems of Homogeneous linear differential equations with constant coefficients constitute an important component of differential equation theory. Their standard fundamental matrix can capture the dynamic behavior and stability of the system. Therefore, solving for the standard fundamental matrix is a critical step in transitioning theoretical analysis and engineering applications. This paper focuses on homogeneous linear differential systems with constant coefficients, aiming to provide a unified expression for all their standard fundamental matrices and to apply this expression to the free vibration problem of simple harmonic oscillators.