Separation of Variables | 分离变量法

📚 Separation of Variables | 分离变量法

Differential equations are the backbone of mathematical modelling in physics, biology, economics, and engineering. Among the many techniques available for solving them, separation of variables stands out as one of the most intuitive and widely used methods for first-order equations. This approach transforms a seemingly complex relationship involving derivatives into a pair of integrals that can be evaluated independently. For IB Mathematics students, mastering separation of variables not only unlocks a range of exam problems but also provides a gateway to understanding real-world phenomena such as population growth, radioactive decay, and cooling processes.

微分方程是物理、生物、经济和工程中数学建模的核心。在众多求解方法中，分离变量法脱颖而出，成为处理一阶方程最直观、应用最广泛的技术之一。这种方法将一个看似复杂的包含导数的关系式转化为一对可以独立计算的积分。对于 IB 数学学生来说，掌握分离变量法不仅能轻松应对考试题，还能打开理解现实世界现象——如种群增长、放射性衰变和冷却过程——的大门。

1. Introduction to Differential Equations | 微分方程概述

A differential equation is any equation that relates a function to its derivatives. In IB Mathematics: Analysis and Approaches HL and Applications and Interpretation HL, the focus is primarily on first-order ordinary differential equations (ODEs) of the form dy/dx = f(x, y). The goal is to find the function y(x) that satisfies this relationship, possibly also satisfying an initial condition such as y(x₀) = y₀.

微分方程是将一个函数与其导数联系起来的方程。在 IB 数学的分析与方法（HL）以及应用与解释（HL）课程中，重点主要放在一阶常微分方程上，形式为 dy/dx = f(x, y)。目标是找到满足这一关系的函数 y(x)，可能需要同时满足一个初始条件，例如 y(x₀) = y₀。

Some differential equations can be solved by direct integration if the right-hand side depends only on x. However, when the expression involves both x and y, we need more advanced techniques. Separation of variables is the simplest non-trivial method for equations where the variables can be algebraically separated onto different sides of the equality.

如果右侧只依赖于 x，有些微分方程可以直接积分求解。但是，当表达式中同时包含 x 和 y 时，就需要更高级的技巧。当方程中的变量可以通过代数方法分离到等号两侧时，分离变量法就是最简单且实用的方法。

2. The Idea of Separable Equations | 可分离方程的概念

An ODE is called separable if it can be written in the form dy/dx = g(x)h(y), where g(x) is a function of x alone and h(y) is a function of y alone. The crucial insight is that we can rearrange the equation so that all terms involving y appear on one side with dy, and all terms involving x appear on the other side with dx. Symbolically, we manipulate the equation into 1/h(y) dy = g(x) dx and then integrate both sides.

如果一个常微分方程可以写成 dy/dx = g(x)h(y) 的形式，其中 g(x) 仅依赖于 x，h(y) 仅依赖于 y，则称该方程是可分离的。关键的洞察在于，我们可以重新排列方程，使得所有涉及 y 的项与 dy 一起放在一边，所有涉及 x 的项与 dx 一起放在另一边。形式上，我们将方程变形为 1/h(y) dy = g(x) dx，然后对两边积分。

The method relies on treating dy/dx not just as a symbol for derivative but as a ratio of differentials, which is justified by the chain rule. This viewpoint is powerful because it reduces the problem of solving a differential equation to evaluating two indefinite integrals, plus handling the constant of integration.

这种方法依赖于将 dy/dx 不仅视为导数的符号，还视为微分的比值，这一做法由链式法则保证其合理性。这种观点之所以强大，是因为它将解微分方程的问题简化为计算两个不定积分，再加上处理积分常数。

3. General Method of Separation of Variables | 分离变量法的一般步骤

The step-by-step procedure can be summarised as follows: first, write the given equation in the form dy/dx = g(x)h(y). Next, separate the variables by rearranging to obtain (1/h(y)) dy = g(x) dx, provided h(y) ≠ 0. Then integrate both sides: ∫(1/h(y)) dy = ∫g(x) dx. Finally, solve for y explicitly if possible, and incorporate any initial condition to find the particular constant of integration. The constant of integration is usually introduced on the right-hand side as C, though sometimes combining constants from both sides is convenient.

具体步骤可以概括如下：首先，将给定方程写成 dy/dx = g(x)h(y) 的形式。接着，通过重排将变量分离，得到 (1/h(y)) dy = g(x) dx，前提是 h(y) ≠ 0。然后对两边积分：∫(1/h(y)) dy = ∫g(x) dx。最后，如果可能的话，显式解出 y，并代入初始条件求出特定的积分常数。积分常数通常引入到右侧记作 C，不过有时将两侧的常数合并会更方便。

In IB exams, finding a general solution in the form y = f(x, C) is often required, and then determining C using y(x₀) = y₀. Sometimes the solution is left implicitly. Students should be comfortable with both explicit and implicit forms.

在 IB 考试中，经常要求求出通解 y = f(x, C)，然后利用 y(x₀) = y₀ 确定 C 的值。有时解也会保留为隐式形式。学生应熟悉显式和隐式两种表达形式。

4. Worked Example: dy/dx = x y | 实例：dy/dx = x y

Consider the simple separable equation dy/dx = xy. Here g(x) = x and h(y) = y. Assuming y ≠ 0, we separate variables to get (1/y) dy = x dx. Integrating both sides yields ln|y| = ½ x² + C. Exponentiating gives |y| = e^(½ x² + C) = e^C · e^(½ x²). We can drop the absolute value by allowing the constant A = ± e^C, so y = A e^(x²/2), where A is an arbitrary nonzero constant. If y = 0 is also a solution, we can include it by letting A = 0.

考虑简单的可分离方程 dy/dx = xy。这里 g(x) = x，h(y) = y。假设 y ≠ 0，分离变量得到 (1/y) dy = x dx。两边积分得 ln|y| = ½ x² + C。取指数后得到 |y| = e^(½ x² + C) = e^C · e^(½ x²)。我们可以通过引入常数 A = ± e^C 去掉绝对值，于是 y = A e^(x²/2)，其中 A 为非零任意常数。如果 y = 0 也是解，我们可以通过允许 A = 0 将其包含在内。

If an initial condition were, say, y(0) = 3, then substituting gives 3 = A e⁰ = A, so the particular solution is y = 3 e^(x²/2). This example illustrates the entire workflow: separate, integrate, handle absolute values using a constant multiplier, and apply the initial condition.

如果给定初始条件，比如 y(0) = 3，代入得 3 = A e⁰ = A，所以特解为 y = 3 e^(x²/2)。这个例子展示了完整的工作流程：分离、积分、用常数乘子处理绝对值、应用初始条件。

5. Initial Conditions and Particular Solutions | 初始条件与特解

A differential equation without initial conditions yields a family of solutions described by a general constant. The initial condition y(x₀) = y₀ picks out one specific member of this family. When solving IB problems, it is essential to substitute the given values immediately after integration (or after obtaining the general solution) to solve for the constant. Sometimes it is algebraically simpler to determine the constant right after the integration step, before exponentiating or simplifying.

没有初始条件时，微分方程给出的是一族由通解常数描述的解。初始条件 y(x₀) = y₀ 选择了这一族解中特定的一个。在解答 IB 题目时，必须在积分后（或得到通解后）立即代入给定值求出常数。有时在积分步骤之后、取指数或化简之前就确定常数，在代数上会更为简便。

For example, in the equation dy/dx = y² cos x with y(π/2) = 1, separating gives y⁻² dy = cos x dx. Integrating, −1/y = sin x + C. Using the condition, −1/1 = sin(π/2) + C → −1 = 1 + C → C = −2. Then −1/y = sin x − 2, so y = 1/(2 − sin x). There is no need to consider multiple branches because the condition determines the sign.

例如，方程 dy/dx = y² cos x，满足 y(π/2) = 1。分离变量得 y⁻² dy = cos x dx。积分：−1/y = sin x + C。利用初始条件：−1/1 = sin(π/2) + C → −1 = 1 + C → C = −2。于是 −1/y = sin x − 2，所以 y = 1/(2 − sin x)。无需考虑多个分支，因为初始条件已确定了符号。

6. Exponential Growth and Decay | 指数增长与衰减

The differential equation dy/dt = k y, where k is a constant, is a fundamental model for natural growth (k > 0) and decay (k < 0). This equation is separable: (1/y) dy = k dt. Integrating gives ln|y| = k t + C, so y = A e^(k t) with A = ± e^C. The constant A equals the initial amount y(0). Thus, the solution is y(t) = y(0) e^(k t).

微分方程 dy/dt = k y，其中 k 为常数，是自然增长（k > 0）和衰减（k < 0）的基本模型。该方程是可分离的：(1/y) dy = k dt。积分得 ln|y| = k t + C，因此 y = A e^(k t)，其中 A = ± e^C。常数 A 等于初始量 y(0)。因此解为 y(t) = y(0) e^(k t)。

In IB contexts, this appears in population dynamics, radioactive decay (with half-life), continuously compounded interest, and more. Students often need to find k from given doubling time or half-life. For exponential decay, k is negative, and the solution can be written as y = y₀ e^(−λ t) with λ > 0.

在 IB 情境中，这出现在种群动力学、放射性衰变（半衰期）、连续复利等。学生常需要根据给定的倍增时间或半衰期求出 k。对于指数衰减，k 为负，解也可写成 y = y₀ e^(−λ t)，其中 λ > 0。

7. Newton’s Law of Cooling | 牛顿冷却定律

Newton’s law of cooling states that the rate of change of the temperature T of an object is proportional to the difference between its temperature and the ambient temperature Tₐ. This gives the separable equation dT/dt = −k (T − Tₐ), where k > 0. Separating variables: 1/(T − Tₐ) dT = −k dt, provided T ≠ Tₐ. Integrating gives ln|T − Tₐ| = −k t + C. Exponentiating, T − Tₐ = A e^(−k t). Applying initial condition T(0) = T₀ yields A = T₀ − Tₐ, so T(t) = Tₐ + (T₀ − Tₐ) e^(−k t).

牛顿冷却定律指出，物体温度 T 的变化率与其温度与环境温度 Tₐ 之差成正比。由此得到可分离方程 dT/dt = −k (T − Tₐ)，其中 k > 0。分离变量：1/(T − Tₐ) dT = −k dt，前提是 T ≠ Tₐ。积分得 ln|T − Tₐ| = −k t + C。取指数：T − Tₐ = A e^(−k t)。利用初始条件 T(0) = T₀ 得 A = T₀ − Tₐ，所以 T(t) = Tₐ + (T₀ − Tₐ) e^(−k t)。

IB problems on cooling often ask to find the time when the object reaches a certain temperature or to determine the constant k from two temperature measurements. Since the absolute value is resolved by knowing whether the object is cooling (T₀ > Tₐ) or heating (T₀ < Tₐ), the same formula works in both cases.

IB 中的冷却问题常要求找出物体达到某一温度所需的时间，或是通过两次温度测量确定常数 k。由于根据物体是在冷却（T₀ > Tₐ）还是加热（T₀ < Tₐ）即可确定绝对值内的符号，同一公式对两种情况都适用。

8. The Logistic Equation (dy/dx = k y (M − y)) | 逻辑斯谛方程

A richer growth model is the logistic differential equation dy/dt = k y (M − y), where M is the carrying capacity. This equation is separable, though the integration requires partial fractions: 1/(y (M − y)) = (1/M)(1/y + 1/(M − y)). Thus, ∫(1/y + 1/(M − y)) dy = ∫ k M dt. Integrating gives ln|y| − ln|M − y| = k M t + C, or ln|y/(M − y)| = k M t + C. Exponentiating produces y/(M − y) = A e^(k M t), where A = e^C. Solving for y gives y = M / (1 + B e^(−k M t)) with B = 1/A, an equivalent form. With an initial condition y(0) = y₀, the constant is determined.

更丰富的增长模型是逻辑斯谛微分方程 dy/dt = k y (M − y)，其中 M 为环境容纳量。该方程是可分离的，但积分需要用部分分式：1/(y (M − y)) = (1/M)(1/y + 1/(M − y))。于是，∫(1/y + 1/(M − y)) dy = ∫ k M dt。积分得 ln|y| − ln|M − y| = k M t + C，即 ln|y/(M − y)| = k M t + C。取指数得 y/(M − y) = A e^(k M t)，其中 A = e^C。解出 y 得 y = M / (1 + B e^(−k M t))，其中 B = 1/A，两者等价。利用初始条件 y(0) = y₀ 可确定常数。

Logistic growth appears in IB AI HL as an example of a more realistic population model. The shape of the solution is an S-curve that levels off at M. The maximum growth rate occurs at y = M/2. Understanding the separation process for this equation reinforces the technique and highlights the importance of algebraic manipulation and partial fractions.

逻辑斯谛增长在 IB AI HL 中作为更现实的人口模型出现。解的形态是一条在 M 处趋于水平的 S 形曲线。最大增长率发生在 y = M/2 处。理解该方程的分离过程可以强化这一技巧，并突显代数操作与部分分式的重要性。

9. Domain Considerations and Absolute Values | 定义域与绝对值处理

When integrating 1/y, we obtain ln|y|. The absolute value ensures the expression is defined for negative y as well. When exponentiating, we often write ± e^C to capture both positive and negative constants. In many IB problems, the context (e.g., population, temperature) dictates that the quantity is positive, so the absolute value can be dropped after verifying the sign. However, showing the step with absolute values and then explaining the sign choice is good practice.

对 1/y 积分时，我们得到 ln|y|。绝对值确保了表达式对负的 y 也有定义。在取指数时，我们常写出 ± e^C，以涵盖正常数和负常数两种情况。在许多 IB 问题中，背景（如人口、温度）表明该量是非负的，因此在验证符号后可以去掉绝对值。不过，展示带绝对值的步骤并解释符号选择是一种良好的习惯。

Additionally, separation of variables is only valid where h(y) ≠ 0. Singular solutions where h(y) = 0 need to be checked separately because they may correspond to equilibrium solutions that are lost during division. For example, in dy/dx = y², y = 0 is a solution, but it might not appear directly from the integrated form 1/y = x + C; it corresponds to the limiting case when |C| → ∞.

此外，分离变量法仅在 h(y) ≠ 0 时成立。h(y) = 0 的奇异解需要单独检查，因为它们可能对应于在除法过程中丢失的平衡解。例如，在 dy/dx = y² 中，y = 0 是一个解，但它可能不会直接从积分后的形式 1/y = x + C 中得到；它对应于 |C| → ∞ 的极限情形。

10. Verifying a Solution Obtained by Separation | 验证分离变量法的解

After finding a candidate solution, it is always wise to verify by differentiation and substitution back into the original differential equation. This is a straightforward process: compute dy/dx from the explicit or implicit solution and check that the original relation holds for all x in the domain. Verification not only confirms correctness but also often catches algebraic mistakes made during integration or when solving for y.

找到候选解后，通过求导并代回原微分方程进行验证总是明智的。这是一个直接的过程：从显式或隐式解中计算出 dy/dx，并检查原关系在定义域内的所有 x 是否成立。验证不仅能确认正确性，还常常能发现在积分或解出 y 的过程中出现的代数错误。

For the equation dy/dx = x y, we found y = A e^(x²/2). Differentiating: dy/dx = A e^(x²/2) · x = x y, which matches the original equation. This step, though simple, gives confidence in the solution and is a useful exam technique.

对于方程 dy/dx = x y，我们得到了 y = A e^(x²/2)。求导：dy/dx = A e^(x²/2) · x = x y，这与原方程一致。这个简单的步骤能增强对解的信心，也是一项有用的考试技巧。

11. Common Pitfalls and Tips | 常见错误与技巧

Students frequently forget the absolute value when integrating 1/y, leading to solutions that are not valid for negative y. Another common mistake is mishandling the constant of integration: it should be introduced on one side only, and care must be taken when exponentiating or applying algebraic manipulations. Also, forgetting to check for division by zero (singular solutions) can lose marks in IB questions where the equilibrium solution is part of the answer.

学生经常在积分 1/y 时忘记绝对值，导致解对负的 y 无效。另一个常见错误是积分常数处理不当：只应在一侧引入常数，并且在取指数或进行代数操作时需要格外小心。此外，忘记检查除以零（奇异解）会在 IB 题目中失分，因为平衡解可能是答案的一部分。

A practical tip is to write the separated form clearly: ‘separate’ implies moving all y-terms to the left with dy and all x-terms to the right with dx. Then immediately write the integral signs. Also, practising with a variety of functions, such as trigonometric functions (e.g., dy/dx = sin x / cos y), exponential functions, and rational expressions, builds confidence.

一个实用技巧是清晰地写出分离后的形式：“分离”意味着将所有 y 项连同 dy 移到左边，所有 x 项连同 dx 移到右边。然后立即写上积分符号。此外，练习各种函数类型，如三角函数（例：dy/dx = sin x / cos y）、指数函数和有理表达式，可以建立信心。

12. Practice Problems and Summary | 练习题与总结

To solidify the technique, here are a few practice exercises: (a) Solve dy/dx = y² eˣ with y(0) = 1. (b) Find the general solution of dy/dx = (y ln x)/x. (c) Solve the initial-value problem dy/dt = 3y(100 − y), y(0) = 10. (d) A liquid cools from 80°C to 60°C in 5 minutes in a room at 20°C. Find the temperature after 10 minutes. Work through these step by step, and verify each solution.

为了巩固技巧，这里提供几道练习题：(a) 解 dy/dx = y² eˣ，y(0) = 1。(b) 求 dy/dx = (y ln x)/x 的通解。(c) 解初值问题 dy/dt = 3y(100 − y)，y(0) = 10。(d) 一杯液体在室温 20°C 下，5 分钟内从 80°C 冷却到 60°C，求 10 分钟后的温度。请逐步操作并验证每一个解。

In summary, separation of variables is a powerful, elegant technique for solving first-order ODEs. By recognising the separable form, separating, integrating, and applying initial conditions, IB students can tackle a wide spectrum of modelling and pure mathematical problems. With careful attention to domain, constants, and verification, this method becomes a reliable tool in any mathematician’s toolkit.

总结而言，分离变量法是求解一阶常微分方程的一种强大而优雅的技术。通过识别可分离形式、分离变量、积分并应用初始条件，IB 学生可以处理广泛的建模和纯数学问题。只要对定义域、常数和验证给予足够重视，这种方法就会成为任何数学工具箱中值得信赖的工具。

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