一类二阶微分方程的解及其稳定性

SOLUTION AND STABILITY OF A CLASS OF SECOND ORDER DIFFERENTIAL EQUATIONS

旨在研究可以刻画弹簧振子振动的一类二阶微分方程的解及其稳定性.首先,从定性的角度应用奇点理论得到微分方程零解的稳定性.然后,从定量的角度利用特征根法和常数变易法得到方程的通解,进一步分析周期解的存在性.最后,利用MATLAB软件绘制该弹簧系统的相图与线素场,直观判断系统零解的稳定性.本文给出了从不同角度判断非齐次微分方程解的稳定性的方法.利用绘制相图与线素场的方法将抽象问题变得直观,为判断微分方程零解的稳定性提供了一种简单易行的新方法.

This article aims to study the solution and stability of a class of second-order differential equations that can describe the vibration of a spring oscillator.From a qualitative point of view,the singularity theory is applied to obtain the stability of the zero solution of the differential equation.From a quantitative point of view,the general solution of the equation is obtained by the characteristic root method and the constant variation method,and the existence of the periodic solution is further analyzed.Finally,MATLAB software is used to draw the phase diagram and line element field of the spring system to intuitively judge the stability of the zero solution of the system.This article gives a method to judge the stability of the solution of inhomogeneous differential equations from different angles.And this article uses the method of drawing the phase diagram and the line element field to make the abstract problem intuitive,and provides a new simple and easy method for intuitively judging the stability of the zero solution of the differential equation.