Differential Equations of the Form dy/dx = f(x)g(y) | 形如 dy/dx = f(x)g(y) 的微分方程

📚 Differential Equations of the Form dy/dx = f(x)g(y) | 形如 dy/dx = f(x)g(y) 的微分方程

Differential equations link a function with its derivative. In AQA A-level Mathematics, a key skill is to solve first-order equations that can be written in the separable form dy/dx = f(x)g(y).

微分方程把一个函数与其导数联系起来。在 AQA A-level 数学中，一项关键技能是求解可写成可分离形式 dy/dx = f(x)g(y) 的一阶方程。

1. What Is a Differential Equation? | 什么是微分方程？

A differential equation is an equation involving an unknown function and one or more of its derivatives. For AQA, the unknown is usually y as a function of x, and the derivative appears as dy/dx.

微分方程是包含未知函数及其一个或多个导数的方程。在 AQA 考试中，未知量通常是 y 关于 x 的函数，导数写成 dy/dx。

The order of a differential equation is the highest derivative that appears. In this topic we focus on first-order equations.

微分方程的阶是出现的最高阶导数。本专题重点研究一阶方程。

dy/dx = 3x² + 2, dy/dx = ky, dy/dx = x sin y

These all describe how the rate of change of y depends on x and/or y.

这些都描述了 y 的变化率如何依赖 x 和/或 y。

2. Recognising Separable Form | 识别可分离变量形式

An equation is separable if the right-hand side can be factored into a function of x only times a function of y only.

如果一个方程的右边可以因式分解为只含 x 的函数乘以只含 y 的函数，则该方程是可分离的。

dy/dx = f(x)g(y)

For example, dy/dx = x²y is separable with f(x)=x² and g(y)=y. The equation dy/dx = x+y is not separable, so it is not solved by this method at A-level.

例如，dy/dx = x²y 是可分离的，其中 f(x)=x²，g(y)=y。方程 dy/dx = x+y 不可分离，因此在 A-level 阶段不用此方法求解。

3. The Separation Method Step by Step | 分离变量法步骤

To solve dy/dx = f(x)g(y):

求解 dy/dx = f(x)g(y) 的步骤：

Step 1: Treat dy/dx as a fraction and divide both sides by g(y), then multiply by dx.
第 1 步：把 dy/dx 当作分式，两边除以 g(y)，再乘以 dx。
Step 2: Arrange so all y terms are on the left with dy and all x terms are on the right with dx.
第 2 步：把所有含 y 的项与 dy 放在左边，所有含 x 的项与 dx 放在右边。
Step 3: Integrate both sides.
第 3 步：两边积分。
Step 4: Add the constant of integration on one side only.
第 4 步：只在一侧加上积分常数。
Step 5: Simplify and, if possible, rearrange to make y the subject.
第 5 步：化简，如有可能，整理成 y 的显式表达式。

∫ 1/g(y) dy = ∫ f(x) dx

Remember that g(y) must not be zero, because division by zero is undefined.

记住 g(y) 不能为零，因为除以零没有定义。

4. Worked Example: Exponential Growth | 例题：指数增长

Solve the differential equation dy/dx = 3y.

求解微分方程 dy/dx = 3y。

First separate the variables: divide both sides by y and multiply by dx:

首先分离变量：两边除以 y，再乘以 dx：

1/y dy = 3 dx

Integrate both sides:

两边积分：

∫ 1/y dy = ∫ 3 dx ⇒ ln|y| = 3x + C

Exponentiate to remove the natural logarithm:

两边取指数以消去自然对数：

|y| = e^(3x+C) = e^C e^(3x)

Let A = e^C. Since A is a positive constant, writing y = Ae^(3x) recovers both positive and negative solutions if A is allowed to be any real non-zero constant.

令 A = e^C。由于 A 是正常数，若允许 A 为任意非零实常数，写成 y = Ae^(3x) 可同时涵盖正负解。

The general solution is therefore:

因此通解为：

y = Ae^(3x)

This is the standard exponential growth model used in population, radioactive decay and finance contexts.

这是用于人口、放射性衰变和金融情境的标准指数增长模型。

5. Worked Example: Trigonometric Separable Equation | 例题：三角函数的可分离方程

Solve dy/dx = x sin y, given that y ≠ nπ.

求解 dy/dx = x sin y，给定 y ≠ nπ。

Separate the variables:

分离变量：

1/sin y dy = x dx

Write 1/sin y as cosec y and integrate:

将 1/sin y 写作 cosec y 并积分：

∫ cosec y dy = ∫ x dx

ln|cosec y – cot y| = ½x² + C

Alternatively, since ∫cosec y dy = ln|tan(y/2)|, the answer may be quoted in that form too. The key AQA skill is separating and integrating correctly.

也可以写成 ∫cosec y dy = ln|tan(y/2)|，答案也可以使用这种形式。AQA 考查的关键是正确分离变量并积分。

For a particular solution, you would substitute a given point, such as (0, π/2), to find C.

对于特解，需代入给定点，例如 (0, π/2)，求出 C。

6. Finding Particular Solutions | 求特解

A general solution contains an arbitrary constant. A particular solution is found by using a boundary condition or initial condition, usually written as y(a)=b.

通解含有一个任意常数。特

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Differential Equations of the Form dy/dx = f(x)g(y) | 形如 dy/dx = f(x)g(y) 的微分方程