AQA数学 微分方程 阶数 线性 分离变量
微分方程（differential equation）是 A-Level 数学中连接代数与应用的桥梁。这篇文章围绕 AQA 考纲中"微分方程的概念：阶数与线性"这一主题，系统讲解什么是微分方程、如何判断阶数与线性、怎样从实际问题建立方程，并给出分离变量法的完整解题流程与考试技巧。
Differential equations are the bridge between algebra and real-world applications in A-Level Mathematics. This article focuses on the AQA syllabus topic "the concept of a differential equation: order and linearity": it explains what a differential equation is, how to identify its order and linearity, how to form equations from real situations, and walks through the full separation-of-variables procedure with exam technique.
1. What Is a Differential Equation 什么是微分方程
微分方程是含有导数（derivative）的方程，它描述的不是某个量本身，而是这个量的变化率与各变量之间的关系。例如 dy/dx = 2x 就是一个最简单的微分方程，它告诉我们函数 y 的变化率等于 2x，解出 y = x² + c 的过程就是求解微分方程。在 AQA 数学中，微分方程通常写作 dy/dx 或 y' 的形式，也可能出现高阶导数 d²y/dx²。理解"方程描述变化"这一本质，是学习微分方程的第一步。
A differential equation is an equation that contains a derivative. Instead of describing a quantity directly, it describes how that quantity changes. For example, dy/dx = 2x is a simple differential equation: it states that the rate of change of y equals 2x, and solving it gives y = x² + c. In AQA A-Level Mathematics, differential equations use notation such as dy/dx, y' or higher derivatives like d²y/dx². Grasping the idea that the equation describes change, rather than the value itself, is the essential first step.
2. The Order of a Differential Equation 微分方程的阶数
微分方程的阶（order）由方程中出现的最高阶导数决定。只含一阶导数 dy/dx 的方程称为一阶微分方程，例如 dy/dx = 3y 和 dy/dx + 2xy = sin x；含有二阶导数 d²y/dx² 的方程称为二阶微分方程，例如 d²y/dx² + 3 dy/dx + 2y = 0。判断阶数时，只需要看方程里最高的导数阶数，与方程中其他项无关。阶数还决定了通解中任意常数的个数：一阶方程的通解含一个任意常数，二阶方程的通解含两个。
The order of a differential equation is determined by the highest derivative it contains. An equation involving only the first derivative dy/dx is called first order, such as dy/dx = 3y or dy/dx + 2xy = sin x. An equation containing the second derivative d²y/dx² is called second order, such as d²y/dx² + 3dy/dx + 2y = 0. To find the order, look only at the highest derivative present; other terms do not matter. The order also tells you how many arbitrary constants the general solution will contain: one for a first-order equation, two for a second-order equation.
3. Linear and Non-linear Differential Equations 线性与非线性微分方程
线性（linear）的含义是：未知函数 y 及其各阶导数都只以一次方出现，且没有 y 与导数的乘积项。一阶线性微分方程可以写成标准形式 dy/dx + P(x)y = Q(x)，例如 dy/dx + 2y = e^x。如果方程中出现 y²、(dy/dx)²、y·dy/dx、sin y 或 e^y 这样的项，方程就是非线性的，例如 (dy/dx)² + y = 0 或 dy/dx = y²。注意，(dy/dx)² 是一阶但非线性，不要把它误认为二阶。线性方程之所以重要，是因为它们满足叠加原理，求解方法也更系统；AQA 考试中绝大多数一阶方程都属于线性或可化为线性的类型。
A differential equation is linear when the unknown function y and all of its derivatives appear only to the first power, with no product terms such as y times dy/dx. A first-order linear equation can be written in the standard form dy/dx + P(x)y = Q(x); for example, dy/dx + 2y = e^x is linear. If any term such as y², (dy/dx)², y·dy/dx, sin y or e^y appears, the equation is non-linear: both (dy/dx)² + y = 0 and dy/dx = y² are non-linear. Note that (dy/dx)² is still first order, just non-linear, so do not mistake it for a second-order equation. Linearity matters because linear equations obey the superposition principle and have systematic solution methods; in AQA exams the vast majority of first-order equations are linear or can be reduced to linear form.
4. General Solutions and Particular Solutions 通解与特解
求解一阶微分方程时，积分会产生一个任意常数 c，带 c 的解族称为通解（general solution），它代表无穷多条曲线。若题目给出初始条件（initial condition），如 x = 1 时 y = 3，代入通解求出 c，就得到特解（particular solution）。以 dy/dx = 2x 为例，通解是 y = x² + c；由 y(1) = 3 得 c = 2，特解为 y = x² + 2。AQA 题目经常分两步：先求通解，再利用条件求特解，漏掉任意常数或忘记代入条件都是常见失分点。
Integrating a first-order differential equation introduces an arbitrary constant c, and the resulting family of solutions is called the general solution; it represents infinitely many curves. When an initial condition is given, such as y = 3 when x = 1, substituting it into the general solution determines c and produces a particular solution. For dy/dx = 2x the general solution is y = x² + c; the condition y(1) = 3 gives c = 2, so the particular solution is y = x² + 2. AQA questions usually ask for the general solution first and then use a condition to find the particular solution. Forgetting the constant or failing to apply the condition are common ways to lose marks.
5. Forming Differential Equations from Real Situations 从实际问题建立微分方程
AQA 常考的一类题是把文字条件翻译成微分方程。关键句式是"变化率与……成正比（proportional to）"：若数量 N 的变化率与 N 成正比，则 dN/dt = kN；若涉及衰减（如放射性衰变、冷却），则 dN/dt = -λN。其他常见模型包括：物体冷却时温度变化率与温差成正比，dθ/dt = -k(θ - T)；容器内液体混合问题，dQ/dt 等于流入量减流出量。建立方程时要注意符号：增长用正比例系数，减少用负号，这正是题目考察概念理解的地方。
A common AQA question type asks you to translate a written situation into a differential equation. The key phrasing is "the rate of change is proportional to": if the rate of change of a quantity N is proportional to N itself, then dN/dt = kN; if the quantity is decaying, such as radioactive decay or cooling, then dN/dt = -λN. Other standard models include Newton's law of cooling, where the rate of change of temperature is proportional to the temperature difference, giving dθ/dt = -k(θ - T), and mixing problems where dQ/dt equals the rate of inflow minus the rate of outflow. Pay attention to signs: growth takes a positive constant, decay takes a negative one, and this is exactly where examiners test conceptual understanding.
6. Separation of Variables 分离变量法
分离变量法是 AQA 求解一阶微分方程的核心方法，适用于可以写成 dy/dx = f(x)·g(y) 形式的方程。步骤：先把含 y 的项移到左边、含 x 的项移到右边，得到 (1/g(y)) dy = f(x) dx；再对两边分别积分，左边对 y 积分、右边对 x 积分；最后整理出 y 的表达式并保留任意常数。例如解 dy/dx = xy：分离得 (1/y)dy = x dx，两边积分得 ln|y| = x²/2 + c，化简为 y = Ae^(x²/2)。若题目给出初始条件，代入即可确定 A。注意积分时每个不定积分都要加常数，但通常合并写成一个 c。
Separation of variables is the core method for solving first-order differential equations in AQA, and it applies whenever the equation can be written as dy/dx = f(x)·g(y). The procedure is: move every y-term to the left and every x-term to the right, giving (1/g(y)) dy = f(x) dx; then integrate both sides separately, the left with respect to y and the right with respect to x; finally rearrange to make y the subject, keeping the arbitrary constant. For example, to solve dy/dx = xy, separate to get (1/y)dy = x dx, integrate to obtain ln|y| = x²/2 + c, and simplify to y = Ae^(x²/2). If an initial condition is supplied, substitute it to fix A. Technically each indefinite integral needs its own constant, but exam convention is to combine them into a single c.
7. Verifying a Solution by Substitution 代入法验证解
验证一个函数是否为微分方程的解，只需把它及其导数代入方程，检查等式两边是否恒等。AQA 的"show that"题就是这样运作的：例如验证 y = e^(2x) 是 dy/dx = 2y 的解，求导得 dy/dx = 2e^(2x)，而 2y = 2e^(2x)，两边相等，验证成立。验证一般解时，可以先对一般解求导，再代入消去常数，得到原方程。这类题分值不高但几乎每年出现，考察的是对"解"的定义的理解，写清求导和代入两步就能拿满分。
To verify that a function is a solution of a differential equation, substitute the function and its derivatives back into the equation and check that both sides are identically equal. This is exactly what AQA "show that" questions require. For example, to verify that y = e^(2x) solves dy/dx = 2y, differentiate to get dy/dx = 2e^(2x); since 2y = 2e^(2x), the two sides match and the verification is complete. When verifying a general solution, differentiate it first, then substitute and eliminate the constant to recover the original equation. These questions carry few marks but appear nearly every year, and they test your understanding of what a solution really is; writing down the differentiation and substitution steps clearly earns full marks.
8. Worked Example: Newton's Law of Cooling 建模实例：牛顿冷却定律
下面是一道完整的 AQA 风格例题。物体初始温度 90°C，置于 20°C 的环境中，冷却速率与温差成正比，比例系数为 k。第一问：建立微分方程。由题意 dθ/dt = -k(θ - 20)。第二问：求通解。分离变量得 dθ/(θ - 20) = -k dt，积分得 ln(θ - 20) = -kt + c，所以 θ = 20 + Ae^(-kt)。第三问：利用初始条件 t = 0 时 θ = 90，得 A = 70，故 θ = 20 + 70e^(-kt)。若已知 5 分钟后温度为 60°C，代入解得 k = -ln(4/7)/5，再求温度降到 40°C 所需时间 t = ln(7/2)/k。整道题完整展示"建模、求解、定解、解释"四步流程，也是考试中 6 到 8 分大题的常见结构。
Here is a complete AQA-style example. An object starts at 90°C in a 20°C room, and its cooling rate is proportional to the temperature difference, with constant k. First, form the differential equation: dθ/dt = -k(θ - 20). Second, find the general solution: separate variables to get dθ/(θ - 20) = -k dt, integrate to obtain ln(θ - 20) = -kt + c, hence θ = 20 + Ae^(-kt). Third, apply the initial condition θ = 90 when t = 0 to find A = 70, giving θ = 20 + 70e^(-kt). If the temperature is 60°C after 5 minutes, substituting gives k = -ln(4/7)/5, and the time to reach 40°C follows from t = ln(7/2)/k. The question shows the full four-step flow of modelling, solving, fixing the constant and interpreting the result, which is the standard structure of a 6 to 8 mark exam question.
9. AQA Exam Technique 考试技巧
AQA 数学 A-Level（7357）共三份试卷，Paper 1 与 Paper 2 各 2 小时、各占 33.3%，微分方程主要出现在纯数部分，常以"建立方程、求解、解释"的分步大题出现，也可能以判断阶数与线性的小题出现。应对策略：第一，把"变化率"翻译成导数时先写清楚 dN/dt 的形式再继续；第二，分离变量后每一步都写出来，积分过程本身就是得分点；第三，积分后立刻写"+ c"，用初始条件求出常数后回代；第四，注意 AQA 允许使用任意科学计算器（包括 CAS），可以用计算器检查数值解，但步骤分只能靠书写过程获得。遇到"show that"题，必须写出完整的中间步骤，只写结论不得分。
AQA A-Level Mathematics (7357) consists of three papers, with Paper 1 and Paper 2 each lasting two hours and each worth 33.3% of the total. Differential equations appear mainly in the pure mathematics content, often as multi-part questions that ask you to form an equation, solve it and interpret the answer, and occasionally as short recall questions on order and linearity. Practical strategies: first, when translating "rate of change" into a derivative, write down the dN/dt form clearly before continuing; second, show every step after separating variables, since the integration working itself earns marks; third, write "+ c" immediately after integrating and substitute the initial condition to fix the constant; fourth, note that AQA allows any scientific calculator including a CAS, so you can check numerical answers, but method marks only come from written working. For "show that" questions, the full intermediate steps must be written down; a bare conclusion earns nothing.
10. Common Mistakes 常见错误
第一个常见错误是把 (dy/dx)² 当成二阶方程：它是一阶非线性方程，判断阶数看最高阶导数，不是看指数。第二个错误是分离变量不彻底，例如把 dy/dx = x + y 直接当成分离型处理，其实它无法分离，需要换思路或题目另有提示。第三个错误是积分后漏掉任意常数，导致通解不完整。第四个错误是处理 ln 时忘记绝对值，或在化简 e^c 时符号出错。第五个错误是代入初始条件后没有回代整理，特解表达式写错。第六个错误是建模时把衰减写成 dN/dt = kN 而漏掉负号。对照这六条检查自己的解题过程，能有效减少失分。
The first common mistake is treating (dy/dx)² as a second-order equation: it is first order and non-linear, because order is decided by the highest derivative, not by any exponent. The second is separating variables incompletely, for example treating dy/dx = x + y as separable when it cannot be separated at all. The third is dropping the arbitrary constant after integrating, leaving an incomplete general solution. The fourth is forgetting the absolute value inside ln, or making a sign error when simplifying e^c. The fifth is failing to substitute back and rearrange after applying the initial condition, so the particular solution is written incorrectly. The sixth is modelling decay as dN/dt = kN without the negative sign. Checking your working against these six items will reduce lost marks considerably.
11. Key Bilingual Terms 关键双语术语
differential equation 微分方程 | order 阶数 | first order 一阶 | second order 二阶 | linear 线性 | non-linear 非线性 | general solution 通解 | particular solution 特解 | arbitrary constant 任意常数 | initial condition 初始条件 | boundary condition 边界条件 | derivative 导数 | rate of change 变化率 | proportional 成正比的 | separation of variables 分离变量法 | integrate 积分 | exponential growth 指数增长 | radioactive decay 放射性衰变 | Newton's law of cooling 牛顿冷却定律 | verify by substitution 代入验证 | modelling 建模 | solution curve 解曲线
These terms cover the vocabulary you need for AQA differential equation questions in both Chinese and English. Practising them in pairs helps you read bilingual exam resources and answer short definition questions confidently.
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