常微分方程中积分因子的性质及其应用
The Properties and Applications of Integrating Factors in Ordinary Differential Equations

积分因子是常微分方程中的一个重要内容，同时也是学生不易掌握和应用的一个知识点。利用积分因子不仅可以求解满足恰当积分条件的一阶微分方程，更可以和微分中值定理产生联系，因此积分因子在各种数学竞赛以及历年考研数学试题中频繁出现。本文在常微分方程教材的基础上，从恰当方程出发，介绍了积分因子以及相关性质。最后，给出了积分因子的一些应用。
The integrating factors are an important content in ordinary differential equations, and also a knowledge point that students find difficult to master and apply. Making use of the integrating factors can not only solve first-order differential equations that satisfy appropriate integration conditions, but also establish connections with the differential mean value theorem. Therefore, the integrating factors have frequently appeared in various mathematical competitions and past postgraduate mathematics exam questions. In this paper, based on the textbook of ordinary differential equations, we introduce integrating factors and related properties from appropriate equations. Finally, some applications of the integrating factors are presented.