常微分方程在中学数学中的应用

Application of Ordinary Differential Equations in Middle School Mathematics

摘要: 在《常微分方程》的学习过程中，我们发现其中的很多思想方法都在中学数学中有着一定的应用，并且很多中学数学问题都可以用微分方程的方法来解答。本篇文章从《常微分方程》教材中借鉴了关于微分方程的部分理论依据，对《常微分方程》的发展历程和它在中学数学中的具体应用进行了总结。重点阐述了物体冷却、水池注水等问题，用微分方程证明欧拉三角公式，借助微分方程研究曲线性质与切线问题，确定基本初等函数的本质特征及常数变易法在中学数学解题中的应用等问题。其中的一些方法技巧对于提高中学数学教师教学能力有一定的指导作用。在文章的正文部分还通过一些详细的例题讲解来让更多的读者了解微分方程及其在中学数学中的具体应用，令广大读者加深对这一方面的理解。

Abstract: In the process of learning “Ordinary Differential Equations”, we can find that many of them have certain applications in middle school mathematics, and many middle school mathematics problems can be solved by differential equations. This paper draws some theoretical basis on differential equations from the textbook “Ordinary Differential Equations” and summarizes the development of “Ordinary Differential Equations” and its specific application in middle school mathematics. It focuses on object cooling, water filling in pools, etc.; proving Euler’s trigonometric formula with differential equations; using differential equations to study the properties of curves and tangent problems; determining the essential characteristics of basic elementary functions and the application of constant variation method in middle school mathematics problem-solving, etc. Some of these methods and skills have a certain guiding role in improving the teaching ability of middle school mathematics teachers. In the main part of the article, some detailed examples are also used to let more readers understand differential equations and their specific applications in middle school mathematics so that readers can deepen their understanding of this aspect.