A-Level数学微分方程分离变量法

1. 微分方程简介 Introduction to Differential Equations

A differential equation is an equation that involves an unknown function and its derivatives. Unlike algebraic equations where the solution is a number, the solution to a differential equation is a function ： a relationship that describes how one quantity changes with respect to another. In pure mathematics, differential equations connect the abstract ideas of calculus (differentiation and integration) to tangible problems：if you know how fast something is changing, can you reconstruct what it is? 微分方程是包含未知函数及其导数的方程。与代数方程不同（代数方程的解是一个数值），微分方程的解是一个函数：它描述了某个量如何随另一个量的变化而变化。在纯数学中，微分方程将微积分（微分和积分）的抽象概念与具体问题联系起来：如果你知道某物变化的速度，你能还原它是什么吗？

In A-Level Mathematics, we focus on first-order ordinary differential equations (ODEs), where the equation contains only the first derivative dy/dx. These equations model a vast range of real-world phenomena：population growth, radioactive decay, Newton’s law of cooling, and the charging and discharging of capacitors. 在A-Level数学中，我们重点学习一阶常微分方程（ODE），即方程中只包含一阶导数 dy/dx。这类方程可以模拟大量现实世界中的现象：人口增长、放射性衰变、牛顿冷却定律以及电容器的充放电过程。

2. 微分方程的分类与阶数 Classification and Order

The order of a differential equation is the highest derivative that appears in it. A first-order ODE contains only dy/dx, while a second-order ODE contains d²y/dx². At A-Level, the syllabus covers only first-order ODEs, and specifically two solution methods：separation of variables and the integrating factor method. 微分方程的阶数是方程中出现的最高阶导数。一阶ODE只包含 dy/dx，而二阶ODE包含 d²y/dx²。在A-Level课程中，只涉及一阶ODE，具体有两个解法：分离变量法和积分因子法。

Differential equations can also be classified as linear or nonlinear. A first-order linear ODE can be written in the standard form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x only. This classification determines which solution method applies：if the equation can be expressed in this linear form, use the integrating factor method; otherwise, check if variables can be separated. Recognising the correct form is often the hardest part of any exam question. 微分方程还可以分为线性和非线性。一阶线性ODE可以写成标准形式 dy/dx + P(x)y = Q(x)，其中 P(x) 和 Q(x) 只是 x 的函数。这一分类决定了适用哪种解法：如果方程能表达为这种线性形式，使用积分因子法；否则检查是否可以分离变量。识别正确的形式往往是任何考试题中最难的部分。

3. 分离变量法：理论 Separation of Variables： Theory

Separation of variables is the simplest method for solving first-order ODEs. It applies when the equation can be rearranged so that all terms involving y appear on one side (with dy) and all terms involving x appear on the other side (with dx). The general form is dy/dx = f(x)g(y), which can be rewritten as (1/g(y)) dy = f(x) dx. 分离变量法是求解一阶ODE最简单的方法。它适用于可以将方程重新整理，使得所有含 y 的项（连同 dy）在一侧，所有含 x 的项（连同 dx）在另一侧的情况。一般形式为 dy/dx = f(x)g(y)，可改写为 (1/g(y)) dy = f(x) dx。

Once the variables are separated, integrate both sides independently. Remember to include the constant of integration (+C) on one side only ： typically the x-side. After integration, rearrange to express y explicitly as a function of x if the question requires it. Use any given initial conditions or boundary values to determine the value of C. 分离变量后，对两侧分别积分。注意只在其中一侧添加积分常数 +C（通常加在 x 侧）。积分后，如果题目要求，重新整理将 y 明确表示为 x 的函数。利用题目给出的初始条件或边界值确定 C 的值。

4. 分离变量法：例题精解 Worked Examples

Example 1： Solve dy/dx = 2xy, given that y = 3 when x = 0. Separate variables： (1/y) dy = 2x dx. Integrate： ln|y| = x² + C. Apply initial condition at x = 0, y = 3： ln 3 = C. Therefore ln|y| = x² + ln 3, which gives y = 3e^(x²). 例题1：求解 dy/dx = 2xy，已知 x = 0 时 y = 3。分离变量：(1/y) dy = 2x dx。积分：ln|y| = x² + C。代入初始条件 x = 0, y = 3：ln 3 = C。因此 ln|y| = x² + ln 3，得 y = 3e^(x²)。

Example 2： The rate of change of a population P with respect to time t is proportional to P. Initially P = 1000, and after 2 hours P = 1500. Find P as a function of t. The ODE is dP/dt = kP. Separate： (1/P) dP = k dt, giving ln P = kt + C, so P = Ae^(kt). Using P(0)=1000 gives A = 1000, and P(2)=1500 gives 1500 = 1000e^(2k), so k = (1/2)ln(1.5). Thus P = 1000e^(t ln(1.5)/2). 例题2：人口 P 关于时间 t 的变化率与 P 成正比。初始时 P = 1000，2小时后 P = 1500。求 P 关于 t 的函数。ODE为 dP/dt = kP。分离变量：(1/P) dP = k dt，得 ln P = kt + C，即 P = Ae^(kt)。利用 P(0)=1000 得 A = 1000，P(2)=1500 得 1500 = 1000e^(2k)，所以 k = (1/2)ln(1.5)。因此 P = 1000e^(t ln(1.5)/2)。

5. 积分因子法：理论 Integrating Factor Method： Theory

When a first-order ODE cannot be separated, but can be written in the linear form dy/dx + P(x)y = Q(x), the integrating factor method applies. The integrating factor (IF) is defined as e^(∫P(x)dx). Multiplying both sides of the equation by the IF transforms the left-hand side into the exact derivative of (y × IF). 当一阶ODE无法分离变量，但可以写成线性形式 dy/dx + P(x)y = Q(x) 时，使用积分因子法。积分因子（IF）定义为 e^(∫P(x)dx)。将方程两边同时乘以IF，左侧可转化为 (y × IF) 的精确导数。

The key insight is that d/dx[y × IF] = IF × dy/dx + y × d(IF)/dx = IF × dy/dx + y × P(x)IF. By construction, this equals IF × Q(x). Therefore y × IF = ∫ IF × Q(x) dx, and the solution is y = (1/IF) × ∫ IF × Q(x) dx. The method converts an apparently unsolvable equation into a straightforward integration problem. 关键洞察在于 d/dx[y × IF] = IF × dy/dx + y × d(IF)/dx = IF × dy/dx + y × P(x)IF。根据构造，这等于 IF × Q(x)。因此 y × IF = ∫ IF × Q(x) dx，解为 y = (1/IF) × ∫ IF × Q(x) dx。该方法将一个看似无法求解的方程转化为直接的积分问题。

6. 积分因子法：例题精解 Worked Examples

Example 3： Solve dy/dx + 2y = e^x. Here P(x) = 2, so IF = e^(∫2 dx) = e^(2x). Multiply through： e^(2x) dy/dx + 2e^(2x) y = e^(2x) e^x = e^(3x). Left side is d/dx[y e^(2x)], so integrate： y e^(2x) = ∫ e^(3x) dx = (1/3)e^(3x) + C. Hence y = (1/3)e^x + Ce^(-2x). 例题3：求解 dy/dx + 2y = e^x。这里 P(x) = 2，所以 IF = e^(∫2 dx) = e^(2x)。两边同乘：e^(2x) dy/dx + 2e^(2x) y = e^(2x) e^x = e^(3x)。左侧是 d/dx[y e^(2x)]，积分得：y e^(2x) = ∫ e^(3x) dx = (1/3)e^(3x) + C。因此 y = (1/3)e^x + Ce^(-2x)。

Example 4： Solve x dy/dx + y = x², given y(1) = 2. First rewrite in standard form： dy/dx + (1/x)y = x. Here P(x) = 1/x, so IF = e^(∫(1/x) dx) = e^(ln x) = x. Multiply： x dy/dx + y = x², which is d/dx[xy] = x². Integrate： xy = x³/3 + C, so y = x²/3 + C/x. Using y(1) = 2： 2 = 1/3 + C, so C = 5/3. Final answer： y = x²/3 + 5/(3x). 例题4：求解 x dy/dx + y = x²，已知 y(1) = 2。首先改写为标准形式：dy/dx + (1/x)y = x。这里 P(x) = 1/x，所以 IF = e^(∫(1/x) dx) = e^(ln x) = x。两边同乘：x dy/dx + y = x²，即 d/dx[xy] = x²。积分：xy = x³/3 + C，所以 y = x²/3 + C/x。代入 y(1) = 2：2 = 1/3 + C，得 C = 5/3。最终答案：y = x²/3 + 5/(3x)。

7. 实际应用建模 Modelling with Differential Equations

Differential equations are the language of change in applied mathematics. In A-Level exam questions, you will often be asked to form a differential equation from a verbal description and then solve it. The key skill is translating proportional relationships into derivative notation：for example, “the rate of cooling is proportional to the temperature difference” becomes dT/dt = -k(T – T_ambient). Watch for keywords like “rate of change”, “directly proportional to”, and “varies with” as clues to form the ODE. 微分方程是应用数学中描述变化的语言。在A-Level考试中，你常常需要根据文字描述建立微分方程然后求解。关键技能是将比例关系转化为导数符号：例如，”冷却速率与温差成正比”转化为 dT/dt = -k(T – T_ambient)。留意”变化率”、”正比于”和”随…变化”等关键词，它们是构建ODE的线索。

Common modelling scenarios include Newton’s law of cooling (exponential decay toward ambient temperature), population models (exponential or logistic growth), mixing problems (concentration of salt in a tank), and chemical reaction rates. In each case, identify the dependent and independent variables, write the rate of change equation, then apply the appropriate solution method. A typical exam problem might read：”A tank contains 100 litres of pure water. Salt solution of concentration 0.2 kg/L flows in at 5 L/min, and the well-mixed solution flows out at 5 L/min. Find the mass of salt S(t) at time t.” The ODE is dS/dt = 1 – S/20, which is linear and solvable by integrating factor. 常见的建模场景包括牛顿冷却定律（向环境温度的指数衰减）、人口模型（指数增长或逻辑斯蒂增长）、混合问题（水箱中盐的浓度）以及化学反应速率。每种情况下，识别因变量和自变量，写出变化率方程，然后应用适当的解法。一道典型的考题可能是：”一个水箱装有100升纯水。浓度为0.2 kg/L的盐溶液以5 L/min的速度流入，充分混合后的溶液以5 L/min的速度流出。求t时刻盐的质量S(t)。” ODE为 dS/dt = 1 – S/20，这是一个线性方程，可用积分因子法求解。

8. 考试技巧与常见错误 Exam Tips and Common Mistakes

The most frequent error in separation of variables is forgetting the absolute value in the logarithm when integrating 1/y. Always write ln|y|, not ln y ： this is essential when y may be negative. Another common mistake is mishandling the constant of integration：write +C on one side only, and remember that e^C is a positive constant, often rewritten as A for neatness. When a question provides boundary conditions, apply them immediately after integration ： do not wait until the very end, as it is easy to lose track of which constants have been determined and which remain unknown. 分离变量法中最常见的错误是在积分 1/y 时忘记对数中的绝对值符号。一定要写 ln|y| 而不是 ln y ：当 y 可能为负时这一点至关重要。另一个常见错误是积分常数的处理不当：只在其中一侧写 +C，并记住 e^C 是一个正常数，通常简化写作 A。当题目给出边界条件时，在积分后立即代入：不要等到最后才处理，因为很容易忘记哪些常数已确定、哪些仍然未知。

For the integrating factor method, students often forget to first rewrite the equation into the standard form dy/dx + P(x)y = Q(x). If the coefficient of dy/dx is not 1, divide through by it before identifying P(x). Also, after finding y, always verify your answer by differentiating and substituting back into the original ODE ： this takes 30 seconds and catches most algebraic errors. When computing the integrating factor e^(∫P(x)dx), remember that you do not need the +C in the exponent：any antiderivative of P(x) works because the constant would be absorbed as a multiplicative factor that cancels out. 对于积分因子法，学生经常忘记先将方程改写为标准形式 dy/dx + P(x)y = Q(x)。如果 dy/dx 的系数不是1，在识别 P(x) 之前先除以该系数。此外，求出 y 后，始终通过求导并代回原ODE来验证你的答案：这只需要30秒，能识别大部分代数错误。在计算积分因子 e^(∫P(x)dx) 时，记住指数中不需要加 +C：P(x) 的任意一个原函数都有效，因为常数会被作为乘法因子吸收并最终抵消。

9. 中英关键术语对照 Key Bilingual Terms

A-Level数学 微分方程 分离变量法