📚 Separable Variables and Homogeneous Differential Equations | 变量分离法与齐次微分方程
In the IB Mathematics Analysis and Approaches HL course, differential equations are a key topic that connects calculus with real-world problems. This article explains two essential techniques: separation of variables and homogeneous differential equations, both of which are regularly tested in IB exams.
在 IB 数学分析与方法（AA）HL 课程中，微分方程是将微积分与现实问题联系起来的关键主题。本文介绍两种基本方法：变量分离法（separation of variables）与齐次微分方程（homogeneous differential equations），这两种方法都是 IB 考试中的常考内容。
1. What Is a Differential Equation? | 什么是微分方程？
A differential equation is an equation that contains at least one derivative, for example dy/dx = 2x. The order of a differential equation is the order of its highest derivative. A solution is a function y(x) that satisfies the equation for all x in its domain. In IB, first-order differential equations are the most common, and they can often be solved by integration.
微分方程是含有至少一个导数的方程，例如 dy/dx = 2x。微分方程的阶数由其最高导数的阶数决定。解是指在其定义域内满足方程的函数 y(x)。在 IB 考试中，一阶微分方程最为常见，通常可以通过积分求解。
There are many types of first-order equations, but the IB syllabus focuses on two particular forms: separable equations and homogeneous equations written as dy/dx = F(y/x). Both require different solution strategies, so it is important to recognise which form you are dealing with.
一阶微分方程有许多类型，但 IB 考纲重点关注两种形式：可分离变量方程，以及可写成 dy/dx = F(y/x) 的齐次方程。两者需要不同的求解策略，因此识别题目属于哪一种形式非常重要。
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