求三阶正系数齐次微分方程的收敛解

摘要

By combining operation with graphics, the convergent solutions of homogeneous differential equation with positive cubic coefficients are analyzed.Three kinds of curves are drawn by discriminant.It is directly judged that the characteristic equation has negative real roots and the solution is convergent when the discriminant is less than or equal to zero, but when the discriminant is more than zero, there is a pair of conjugate compound roots.When the real part of the root is greater than or equal to zero, the solution is divergent.In view of this situation, it is known that two undetermined constants must be tending to zero.Relevantly, we discuss how to find the general solution and convergent solution of homogeneous equation, and modify the normal statement when the discriminant equals zero in mathematical formula.%采用将操作与图形相结合的方法, 分析了三次正系数齐次微分方程的收敛解, 分别用判别式大于零、等于零、小于零的不同状态绘制三种曲线图, 得到判别式小于等于零时特征方程具有负实根并且解是收敛的.但是当判别式大于零时, 有一对共轭的复合根, 当根的实部大于等于零时, 解是发散的.由此可知, 两个未经验证的常数必须趋于零.同时, 讨论了如何找到齐次方程的通解和收敛解, 并将数学公式中判别式等于零的状态做了重新论述.