Separable Differential Equations | 微分方程中的变量分离法

📚 Separable Differential Equations | 微分方程中的变量分离法

Separable differential equations are one of the most important and accessible techniques for solving first-order differential equations. In the IB Mathematics curriculum, students encounter equations of the form dy/dx = g(x)h(y), where the rate of change depends on x and y through a product of a function of x and a function of y. The key idea is to separate the variables so that all y-terms are on one side and all x-terms on the other, then integrate both sides.

可分离变量的微分方程是求解一阶微分方程中最重要也最容易掌握的方法之一。IB 数学课程中，学生需要处理形如 dy/dx = g(x)h(y) 的方程，其中变化率由 x 的函数与 y 的函数的乘积决定。核心思想是分离变量，使所有含 y 的项在一边，所有含 x 的项在另一边，然后两边分别积分。

1. What Are Separable Differential Equations? | 什么是可分离变量的微分方程

A first-order differential equation is called separable if it can be written in the form dy/dx = g(x)h(y). Here g(x) depends only on x and h(y) depends only on y. Many physical laws, such as Newton’s law of cooling, radioactive decay, and population growth, can be modelled by separable equations.

一个一阶微分方程如果能够写成 dy/dx = g(x)h(y) 的形式，就称为可分离变量的微分方程。其中 g(x) 只依赖于 x，h(y) 只依赖于 y。许多物理定律，如牛顿冷却定律、放射性衰变和人口增长，都可以用可分离变量的方程来建模。

For example, dy/dx = 2xy is separable because it can be written as dy/dx = (2x)(y). On the other hand, dy/dx = x + y is not separable directly, because the right-hand side cannot be split into a product of a pure x-function and a pure y-function.

例如，dy/dx = 2xy 是可分离的，因为它可以写成 dy/dx = (2x)(y)。而 dy/dx = x + y 不能直接分离，因为右侧无法拆成一个纯 x 函数与一个纯 y 函数的乘积。

2. The General Method | 一般求解步骤

Suppose we have a separable equation dy/dx = g(x)h(y), with h(y) ≠ 0. We can treat dy/dx as a ratio of differentials and rearrange:

设有可分离方程 dy/dx = g(x)h(y)，其中 h(y) ≠ 0。我们可以将 dy/dx 看作微分的比值并重新整理：

1/h(y) dy = g(x) dx

Then integrate both sides:

然后两边积分：

∫ 1/h(y) dy = ∫ g(x) dx + C

The constant C is the constant of integration. After evaluating the integrals, we obtain a relation between x and y. If possible, solve this relation explicitly for y.

常数 C 是积分常数。计算积分后，我们得到 x 与 y 的一个关系式。如果可能，再将其显式解出 y。

3. Worked Example 1: Basic Separation | 例题1：基础分离变量

Solve the differential equation dy/dx = x/y, where y ≠ 0.

求解微分方程 dy/dx =

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