Differential Equations: Separation of Variables | 微分方程：变量分离法

📚 Differential Equations: Separation of Variables | 微分方程：变量分离法

n the Edexcel Pure Mathematics 3 specification, the method of separation of variables is the primary technique for solving first-order ordinary differential equations. This article, inspired by the revision material on page 142 of pdfjoiner_(4), provides a thorough walkthrough of the concept, worked examples, and essential exam tips. By mastering this method, you will be well-equipped to tackle one of the most predictable and rewarding question types on your paper.

微分方程是 A-Level 数学的基石，使我们能够建立从人口增长到放射性衰变的各种模型。在爱德思纯数学 3 的考纲中，变量分离法是求解一阶常微分方程的主要技术。本文基于 pdfjoiner_(4) 第 142 页的复习资料，对这一概念进行了透彻的讲解，并提供详细例题和重要的应试技巧。掌握该方法后，你将能从容应对试卷中最常见且最有把握的题型之一。

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

first-order equations where the highest derivative is the first derivative. For example, dy/dx = 3x² is a simple differential equation whose general solution is found by direct integration: y = x³ + C, where C is an arbitrary constant. However, in many cases the derivative is mixed with the dependent variable y, making direct integration impossible without first separating the variables.

微分方程是包含导数（如 dy/dx 或 d²y/dx²）的方程。在爱德思 P3 的范畴中，你主要处理的是最高阶导数为一阶导数的方程。例如，dy/dx = 3x² 就是一个简单的微分方程，通过直接积分可求得通解 y = x³ + C，其中 C 为任意常数。然而，在许多情形下导数与因变量 y 混合在一起，若不先将变量分离，就无法直接积分。

(y), where the right-hand side is a product of a function of x and a function of y. This is the core structure we will explore. The ability to recognise this form and rearrange it correctly is the first skill you must develop. Without this step, integration cannot proceed smoothly.

需要分离变量的一阶微分方程经典形式为 dy/dx = f(x) g(y)，其右侧是 x 的函数与 y 的函数的乘积。这是我们即将探讨的核心结构。能够识别这种形式并正确地重新排列，是你必须培养的第一项技能。没有这一步骤，积分便无法顺利进行。

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2. Introducing Separation of Variables | 分离变量法简介

iables technique relies on treating the derivative dy/dx as a fraction, which allows us to 'move' s (1/g(y)) dy = f(x) dx. Then we integrate both sides independently to obtain an implicit or explicit solution for y. This method is valid provided g(y) ≠ 0, a condition that may need to be considered when handling boundary conditions.

变量分离法依赖于将导数 dy/dx 视为分式的思想，这使得我们可以把所有含 y 的项“移”到一边，把所有含 x 的项移到另一边。具体而言，若给出的方程为 dy/dx = f(x) g(y)，可将其改写为 (1/g(y)) dy = f(x) dx。然后对两边分别积分，得到 y 的隐式解或显式解。这种方法在 g(y) ≠ 0 的前提下有效，处理边界条件时可能需要考虑这一限制。

It is helpful to contrast this with equations that can be solved by direct integration alone. The table below highlights the key differences between the two types you will encounter in Edexcel examinations.

将其与仅靠直接积分就能求解的方程进行对比会很有帮助。下表列出了你在爱德思考试中会遇到的这两类方程之间的主要区别。

Type of Equation	Example	Method
Direct Integration	dy/dx = 6x - 2	Integrate directly with respect to x: y = 3x² - 2x + C
Separation of Variables	dy/dx = x y²	Rewrite as (1/y²) dy = x dx, then integrate both sides

Separable equations often arise in real-world modelling, making this technique particularly valuable beyond the exam room. The key insight is that once the variables are separated, the problem reduces to two standard integrals, which you can evaluate using the integration skills developed throughout the P3 course.

可分离方程经常出现在现实世界的建模中，因此该技巧在考场之外同样极具价值。关键的洞见在于：一旦变量被分离，问题就简化为两个标准积分，你可以运用在 P3 课程中培养的积分技能来求解。

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3. The Step-by-Step Method | 方法步骤

To solve a differential equation by separation of variables, follow this systematic procedure. First, rewrite the equation so that all terms containing y (including dy) are on one side and all terms containing x (including dx) are on the other. This often involves dividing by g(y) and multiplying by dx. Second, integrate both sides of the resulting equation. Remember to include the constant of integration on one side only; adding it to the right-hand side is standard, but consistency is key. Third, simplify the integrated expression as much as possible, applying any given initial conditions to find the particular solution. Finally, if the question asks for an explicit solution, rearrange to make y the subject.

要用变量分离法求解微分方程，请遵循以下系统步骤。首先，改写方程，使所有含 y 的项（包括 dy）位于等式一侧，所有含 x 的项（包括 dx）位于另一侧。这通常需要除以 g(y) 并乘以 dx。其次，对所得方程的两边进行积分。务必只在一边加上积分常数；习惯上将其加在右侧，但保持一致性至关重要。第三，尽可能化简积分后的表达式，并利用给定的初始条件求出特解。最后，如果题目要求显式解，再通过移项将 y 写成显函数形式。

Let us formalise the core integral relationship. If dy/dx = f(x) g(y), then we obtain:

∫ (1/g(y)) dy = ∫ f(x) dx + C

When integrating 1/g(y), you may need to use standard results such as ∫ 1/y dy = ln|y| tant C is determined using a boundary condition, typically given as a point (x₀, y₀) that lies on the solution curve.

让我们正式地表述这个核心积分关系。若 dy/dx = f(x) g(y)，则可得到：

∫ (1/g(y)) dy = ∫ f(x) dx + C

在对 1/g(y) 进行积分时，你可能需要用到标准结果，例如 ∫ 1/y dy = ln|y|，或其他幂次下的 ∫ yⁿ dy。除非上下文确定 y 为正，否则应始终在自然对数中考虑绝对值。常数 C 可通过边界条件求得，该条件通常以位于解曲线上的点 (x₀, y₀) 的形式给出。

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4. Worked Example 1: Exponential Relationship | 示例 1：指数关系

ltiplying by dx, obtaining (1/y) dy = 2x dx. Next, integrate both sides: ∫ (1/y) dy = ∫ 2x dx, which yields ln|y| = x² + C. Since we are given that y = 5 when x = 0, we can determine C: ln|5| = 0 + C, so C = ln 5. Thus the particular solution is ln|y| = x² + ln 5, or equivalently, by exponentiating, y = 5 e^x². Because y is positive from the initial condition, we can drop the absolute value signs.

考虑微分方程 dy/dx = 2x y，初始条件为 y(0) = 5。首先分离变量：两边除以 y 并乘以 dx，得到 (1/y) dy = 2x dx。接着，对两边积分：∫ (1/y) dy = ∫ 2x dx，得到 ln|y| = x² + C。由于已知当 x = 0 时 y = 5，可求出 C：ln|5| = 0 + C，因此 C = ln 5。于是特解为 ln|y| = x² + ln 5，或等价地，通过指数化得到 y = 5 e^x²。由于初始条件可知 y 为正，可以去掉绝对值符号。

rtable manipulating logarithmic forms and recognising that a sum of logarithms, such as ln|y| = x² + ln 5, can be combined into ln|y/5| = x², which directly gives y = 5 e^x². Careful step-by-step working minimises arithmetic errors.

该示例展示了爱德思考题中一种非常常见的模式：所得解为指数函数。你应该能熟练处理对数形式，并认识到对数之和（如 ln|y| = x² + ln 5）可以合并为 ln|y/5| = x²，从而直接得到 y = 5 e^x²。细致地逐步求解可最大限度地减少算术错误。

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5. Worked Example 2: Trigonometric Context | 示例 2：三角函数背景

given that y(π/2) = 3. Separate variables to get (1/y) dy = cos x dx. Integration gives ln|y| bstituting x = π/2 and y = 3 yields ln 3 = sin(π/2) + C = 1 + C, so C = ln 3 - 1. Hence ln|y| = sin x + ln 3 - 1. Exponentiating: y = 3 e^{sin x - 1}, which simplifies to y = (3/e) e^{sin x}. Since the initial y is positive, we write y = 3 e^{sin x - 1}. This question could easily appear in a P3 paper where trigonometric functions feature heavily.

现在尝试一个三角函数示例：求解 dy/dx = y cos x，已知 y(π/2) = 3。分离变量得 (1/y) dy = cos x dx。积分得 ln|y| = sin x + C。代入 x = π/2 和 y = 3，得到 ln 3 = sin(π/2) + C = 1 + C，所以 C = ln 3 - 1。因此 ln|y| = sin x + ln 3 - 1。指数化得 y = 3 e^{sin x - 1}，简化后为 y = (3/e) e^{sin x}。由于初始 y 为正，我们写作 y = 3 e^{sin x - 1}。这类题目很可能出现在三角函数占重要地位的 P3 试卷中。

Notice how the integral of cos x is sin x, a standard result that you must know instantly. The manipulation of the constant when exponentiating is another area where marks are easily lost. Always show the explicit evaluation of C before rearranging, as examiners expect to see the substitution step clearly. Failure to do so can result in the loss of method marks, even if the final answer is correct.

注意，cos x 的积分是 sin x，这是你必须烂熟于心的标准结果。在指数化时对常数的处理是另一个容易失分的环节。在重新排列之前，务必展示对 C 的明确求解过程，因为考官期望看到清晰的代入步骤。若未能做到这一点，即便最终答案正确，也可能失去方法分。

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6. Applications: Growth and Decay Models | 应用：增长与衰减模型

stant. If k > 0, we have exponential growth; if k < 0, exponential decay. Using separation of variables, we obtain y = A e^kt 0. This model appears in population dynamics, radioactive decay, and continuously compounded interest, and regularly features in Edexcel exam questions that blend differentiation with integration.

在应用情境中，一个量的变化率常与该量自身成正比，从而形成方程 dy/dt = k y，其中 k 为常数。若 k > 0，表现为指数增长；若 k < 0，表现为指数衰减。通过变量分离法，我们得到 y = A e^kt，其中 A 为 t = 0 时的初始量。该模型出现在人口动力学、放射性衰变和连续复利计算中，并经常出现在爱德思考试中，将微分与积分结合在一起考查。

ber of bacteria N satisfies dN/dt = 0.2 N, with N = 100 at t = 0. You would find N = 100 e^0.2t, and then use it to calculate the time taken for the population to double. This requires solving e^0.2t multi-step problems test your algebraic fluency and your understanding of the underlying differential model.

例如，一个典型的考题可能会给出细菌数量 N 满足 dN/dt = 0.2 N，且在 t = 0 时 N = 100。你将求出 N = 100 e^0.2t，然后利用它计算种群数量翻倍所需的时间。这需要求解 e^0.2t = 2，因此 t = (ln 2)/0.2 ≈ 3.47 小时。这类多步骤问题既考验你的代数熟练度，也考察你对底层微分模型的理解。

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7. Common Mistakes and How to Avoid Them | 常见错误及避免方法

rgetting to include the constant of integration immediately after integrating. Without the '+ C' ling the absolute value in the logarithm. In many pure maths problems, you should write ln|y| t could be zero; a complete solution might require a separate constant solution if g(y) = 0 leads to a valid solution.

最常见的错误之一是在积分之后忘记立即加上积分常数。缺少“+ C”的解是不完整的，将会失分。另一个常见的陷阱是错误处理对数中的绝对值。在许多纯数学问题中，你应该写 ln|y| 而不是 ln y，除非给定条件保证了正值。此外，许多学生在除以 y 的函数时，没有检查它是否可能为零；若 g(y) = 0 也构成有效解，完整的解答可能需要单独给出一个常数解。

The list below summarises the key checks you should perform when solving a separable differential equation:

• Did I separate all y terms onto the dy side and all x terms onto the dx side?
• Did I integrate both sides correctly, including the constant of integration?
• Have I applied the initial condition accurately to evaluate C?
• Is my final answer in the form requested (explicit vs. implicit)?
• Have I considered any restrictions, such as y ≠ 0 or the domain of x?

下面的清单总结了求解可分离微分方程时应进行的关键检查：

• 我是否已将所有含 y 的项分离到 dy 一侧，所有含 x 的项分离到 dx 一侧？
• 我是否正确地对两边进行了积分，并包含了积分常数？
• 我是否准确地应用了初始条件来求出 C？
• 最终答案是否符合所要求的形式（显式或隐式）？
• 我是否考虑了任何限制条件，例如 y ≠ 0 或 x 的定义域？

By systematically incorporating these checks, you will avoid careless mistakes and present a polished, examiner-friendly solution. Practice with past papers will reinforce these habits until they become second nature.

通过系统地进行这些检查，你将避免粗心错误，并呈上令考官满意的严谨解答。借助历年真题的练习，这些习惯会得到强化，直至成为你的第二天性。

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8. Exam Tips for Edexcel P3 | 爱德思 P3 考试技巧

em. The method is often tested alongside integration techniques such as ∫ a/(bx) dx, or requiring the use of partial fractions after separation. You may be given a differential equation as part of a rates-of-change context that then asks for a particular solution and a subsequent evaluation. Read the question carefully to determine whether the solution should be left in implicit form or rearranged.

在爱德思纯数学 3 中，变量分离法通常以一道 6 到 8 分的独立题目出现，或是更大建模问题的一部分。该方法常常与积分技巧一起考查，例如 ∫ a/(bx) dx，或者在分离后需要使用部分分式。你可能会在变化率背景中得到一个微分方程，然后要求求出特解并进行后续计算。仔细读题，判断解答应以隐式形式保留，还是需要重新整理。

When tackling such problems, it is wise to use a structured layout. Write 'Separating variables:' followed by the rearranged equation. Then 'Integrating both sides:' and show the integrals. Next, 'Using the condition...' to find C. Finally, clearly state the particular solution. Neat presentation can earn method marks even if a minor arithmetic slip occurs later. Time management is also crucial; if you are confident with separation of variables, you can secure these marks quickly and move on to more challenging questions.

在解决这类问题时，采用结构化书写是明智的。写下“分离变量：”，然后给出重排后的方程。接着写“两边积分：”，并展示积分结果。然后，“利用条件……”求出 C。最后，清晰地陈述特解。即使后面出现微小的算术错误，整洁的呈现方式仍可为你赢得方法分。时间管理同样至关重要；若你对变量分离法充满信心，就能快速拿下这些分数，然后转向更具挑战性的题目。

Remember that this topic is synoptic, often calling on integration of trigonometric, exponential, and rational functions. A solid revision of P3 integration will directly boost your performance here. Review standard integrals, logarithmic forms, and the handling of coefficients. Separation of variables is one of the most scoring topics if approached methodically.

请记住，该主题是融会贯通的，经常要求对三角函数、指数函数和有理函数进行积分。扎实复习 P3 的积分内容将直接提升你在此处的表现。重温标准积分、对数形式以及系数的处理。只要方法得当，变量分离法是得分率最高的主题之一。

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