📚 AS Mathematics: Differential Equations Essential Guide | AS 数学：微分方程 考点精讲
Differential equations are a fundamental topic in AS Mathematics, bridging pure calculus and real‑world modelling. They describe how quantities change in relation to one another, and mastering the skills to form and solve them is vital for exam success. This guide unpacks the essential concepts, step‑by‑step methods, common applications, and typical pitfalls.
微分方程是 AS 数学的基础课题，连接了纯微积分与现实世界的建模。它们描述了量如何随另一个量变化而变化，掌握建立和求解这些方程的核心技能对考试成功至关重要。本指南将系统梳理基本概念、分步解题方法、常见应用及典型误区。
1. What Is a Differential Equation? | 什么是微分方程？
A differential equation is an equation that contains an unknown function and one or more of its derivatives. In AS Mathematics, we mainly work with first‑order ordinary differential equations, where the highest derivative is the first derivative dy/dx or f'(x).
微分方程是一个包含未知函数及其一个或多个导数的等式。在 AS 数学中，我们主要研究一阶常微分方程，即最高阶导数为 dy/dx 或 f'(x) 的方程。
le number but a family of functions, such as y = x³ + C, where C is an arbitrary constant.
例如，dy/dx = 3x² 就是一个微分方程。它的解不是单一数值，而是一族函数，例如 y = x³ + C，其中 C 为任意常数。
ion is found when an initial condition (e.g., y(2) = 5) is used to determine the constant.
通解包含一个任意常数；当利用初始条件（如 y(2) = 5）确定常数后，即可得到特解。
2. Forming Differential Equations | 构建微分方程
ons ask you to form a differential equation from a worded description. Key phrases such as ‘the rate of change of y is proportional to x’ translate directly to dy/dx ∝ x, or dy/dx = kx, where k is a constant.
许多考题要求根据文字描述构建微分方程。关键表述如“y 的变化率正比于 x”可直接翻译为 dy/dx ∝ x，即 dy/dx = kx，其中 k 是常数。
Look out for relationships involving ‘inversely proportional’, ‘jointly proportional’, or ‘net rate’ as the difference between inflow and outflow rates. Always introduce a constant of proportionality k.
注意“反比”、“联合比例”或“净速率”（流入与流出速率之差）等关系。务必引入比例常数 k。
Example: ‘The population P grows at a rate proportional to its current size’ → dP/dt = kP. ‘ ls at a rate proportional to the difference between T and room temperature 20°C’ → dT/dt = –k(T – 20).
示例：“种群数量 P 的增长速率与其当前规模成正比” → dP/dt = kP。“温度 T 下降的速率与 T 和室温 20°C 的差值成正比” → dT/dt = –k(T – 20)。
3. Separable Differential Equations | 可分离变量的微分方程
A first‑ ion is called separable if it can be written in the form g(y) dy = f(x) dx. That is, all y‑terms (including dy) can be placed on one side and all x‑terms (including dx) on the other.
如果一个一阶微分方程可以写成 g(y) dy = f(x) dx 的形式，就称为可分离变量的方程。也就是说，所有含 y 的项（包括 dy）可移到一边，所有含 x 的项（包括 dx）移到另一边。
rated, the equation is solved by integrating both sides separately: ∫ g(y) dy = ∫ f(x) dx.
这是 AS 考试中最常见的微分方程类型。分离变量后，通过两边分别积分求解：∫ g(y) dy = ∫ f(x) dx。
Always remember to include a constant of integration on one side, usually the right‑hand side. The constant appears as +C.
务必记住在一边（通常是右边）加上积分常数，写为 +C。
4. Solving Separable Equations Step by Step | 逐步求解可分离方程
Let's solve dy/dx = (2x)/(y), with initial condition y(0) = 4.
我们来求解 dy/dx = (2x)/(y)，初始条件为 y(0) = 4。
Step 1: Separate variables. Rewrite as y dy = 2x dx.
第一步：分离变量。重写为 y dy = 2x dx。
Step 2: Integrate both sides. ∫ y dy = ∫ 2x dx → (1/2)y² = x² + C.
第二步：两边积分。∫ y dy = ∫ 2x dx → (1/2)y² = x² + C。
Step 3: Apply initial condition. When x = 0, y = 4 → (1/2)(16) = 0 + C → C = 8.
第三步：代入初始条件。当 x = 0 时 y = 4 → (1/2)(16) = 0 + C → C = 8。
Step 4: Rearrange for y explicitly if possible. (1/2)y² = x² + 8 → y² = 2x² + 16 → y = √2x² + 16. Since y(0) > 0, we take the positive square root.
第四步：尽可能将 y 显式化。(1/2)y² = x² + 8 → y² = 2x² + 16 → y = √2x² + 16。由于 y(0) > 0，取正平方根。
In some cases, the solution may be left in implicit form, e.g., y² – 2x² = 16, if the question does not require an explicit expression.
若题目不要求显式表达式，可保留隐式解，例如 y² – 2x² = 16。
5. Particular Solutions and Initial Conditions | 特解与初始条件
need an initial condition, usually given as a pair (x₀, y₀) such that when x = x₀, y = y₀.
通解含有一个任意常数 C。要得到特解，需要给定一个初始条件，通常以 (x₀, y₀) 的形式给出，即当 x = x₀ 时 y = y₀。
After finding C, substitute back into the general solution. Always check that the particular solution satisfies the original differential equation and the initial condition.
求出 C 后，代回到通解中。务必检查特解是否满足原微分方程和初始条件。
If the solution is in implicit form, you can still determine C by substituting the initial values directly into the integrated equation.
如果解为隐式形式，仍可直接将初始值代入积分后的等式来求出 C。
Be careful: sometimes the initial condition helps choose the correct sign when taking square roots, as seen in the previous example.
请注意：有时初始条件有助于在取平方根时选择正确的符号，如上一示例所示。
6. Differential Equations in Context: Growth and Decay | 应用背景：增长与衰减
decay (k negative), or continuously compounded interest. The general solution is P = P₀ ekt, where P₀ is the initial amount when t = 0.
自然增长模型 dP/dt = kP 用于描述种群增长、放射性衰变（k 为负）或连续复利。其通解为 P = P₀ ekt，其中 P₀ 是 t = 0 时的初始量。
To solve, separate variables: dP/P = k dt → ∫ dP/P = ∫ k dt → ln|P| = kt + c → P = ekt+c = Aekt. Setting t = 0 gives A = P₀.
求解时分离变量：dP/P = k dt → ∫ dP/P = ∫ k dt → ln|P| = kt + c → P = ekt+c = Aekt。令 t = 0 得 A = P₀。
For decay, the half‑life T is the time taken for half the substance to decay: when P = P₀/2, ekT = 1/2 → kT = ln(1/2) = –ln 2, so T = (ln 2)/(–k).
对于衰变，半衰期 T 是指衰减到一半所花的时间：当 P = P₀/2 时，ekT = 1/2 → kT = ln(1/2) = –ln 2，所以 T = (ln 2)/(–k)。
These models appear regularly in applied questions. Always define your variables clearly and state any assumptions (e.g., unlimited resources for growth).
这些模型经常出现在应用题中。务必清楚定义变量，并说明假设（例如增长不受资源限制）。
7. Newton's Law of Cooling | 牛顿冷却定律
Newton' onal to the difference between its own temperature and the ambient temperature Tₐ: dT/dt = –k(T – Tₐ).
牛顿冷却定律指出，物体的温度变化速率与其自身温度和环境温度 Tₐ 的差值成正比：dT/dt = –k(T – Tₐ)。
This is a separable equation. Separate: dT/(T – Tₐ) = –k dt. Integrate: ln|T – Tₐ| = –kt + C. Then T – Tₐ = Ae–kt, so T = Tₐ + Ae–kt.
这是一个可分离变量的方程。分离变量：dT/(T – Tₐ) = –k dt。积分得：ln|T – Tₐ| = –kt + C。于是 T – Tₐ = Ae–kt，所以 T = Tₐ + Ae–kt。
Using initial condition T(0) = T₀ gives A = T₀ – Tₐ. The temperature approaches Tₐ as t → ∞.
利用初始条件 T(0) = T₀ 得 A = T₀ – Tₐ。当 t → ∞ 时，温度趋近于 Tₐ。
Exam questions often provide two temperatures at two different times to find k first, then predict a future temperature or time.
考试题目通常会给出两个不同时刻的温度，先求出 k，然后预测未来温度或所需时间。
8. Slope Fields and Graphical Solutions | 斜率场与图解
A slope field (or direction field) is a graphical representation of a first‑ f(x,y). Short line segments are drawn at grid points, each with slope f(x,y) at that point.
斜率场（或称方向场）是一阶微分方程 dy/dx = f(x,y) 的图形表示。在网格点处画出短线段，每一线段的斜率等于该点处 f(x,y) 的值。
By following the direction of the segments, you can sketch approximate solution curves without actually solving the equation analytically.
沿着线段的方向，你可以勾勒出近似的解曲线，而无须真正解析求解方程。
AS exams may ask you to match a slope field with its differential equation, or to sketch a solution passing through a given point. Pay attention to where the slope is zero (horizontal segments) or undefined (vertical segments).
AS 考试可能要求你将斜率场与相应的微分方程匹配，或者画出经过给定点的解曲线。注意斜率为零（水平线段）或未定义（垂直线段）的位置。
tions occur when dy/dx = 0 for all x; these appear as horizontal lines in the slope field.
平衡解出现在对所有 x 都有 dy/dx = 0 的情况，在斜率场中表现为水平直线。
9. Checking Solutions | 验证解
A quick way to verify a proposed solution is to differentiate it and substitute back into the original differential equation. The left‑hand side should equal the right‑hand side identically.
验证所求解的一种快速方法是求导并代入原微分方程。左侧与右侧应恒等。
For example, to check that y = 2e3x is a solution of dy/dx = 3y, differentiate: dy/dx = 6e3x = 3(2e3x) = 3y, so it is a solution.
例如，要验证 y = 2e3x 是 dy/dx = 3y 的解，求导：dy/dx = 6e3x = 3(2e3x) = 3y，因此它是一个解。
This method is also useful when you are given a family of functions and need to show that they satisfy a given differential equation.
当题目给定一族函数并要求证明它们满足某个微分方程时，这种方法也很管用。
Always check initial conditions too: plug in the given x₀, y₀ and verify that your particular solution satisfies them.
也要始终检查初始条件：代入给定的 x₀、y₀，验证你的特解是否满足它们。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Forgetting the constant of integration. Always add +C immediately after integrating, not later.
错误一：忘记积分常数。务必在积分后立即加上 +C，而不是之后再加。
Mistake 2: Mixing up the sides when separating variables. Carefully multiply or divide to isolate dy with y‑terms and dx with x‑terms.
错误二：分离变量时混淆两边。仔细乘除，使 dy 与含 y 项在一起，dx 与含 x 项在一起。
Mistake 3: Incorrectly handling absolute values in logarithms. Remember ∫ dy/y = ln|y| e value can usually be dropped after exponentiating because the constant absorbs the sign.
错误三：错误处理对数中的绝对值。记住 ∫ dy/y = ln|y|，但在取指数后绝对值通常可去掉，因为常数会吸收符号。
Exam tip: Show all steps clearly: ‘Separate variables’, ‘Integrate both sides’, ‘Use initial condition’. Method marks are awarded for these steps even if the final answer is wrong.
考试技巧：清晰写出所有步骤：“分离变量”、“两边积分”、“代入初始条件”。即使最终答案错误，这些步骤仍可获得方法分。
Exam tip: When a context question asks for the time to reach a certain value, substitute into the solution, isolate the exponential, and take natural logarithms. Simplify using log rules where possible.
考试技巧：当背景题要求计算达到某一值所需时间时，代入解，分离指数项，然后取自然对数。尽可能使用对数律化简。
Exam tip: If the question states ‘ you must rearrange the integrated equation explicitly. Otherwise, an implicit form is often acceptable.
考试技巧：如果题目要求“以 y = f(x) 的形式给出答案”，你必须将积分后的方程显式化。否则，隐式形式通常可接受。
Exam tip: ertical segments where the expression is undefined. Match these key features to equations.
考试技巧：对于斜率场，当 f(x,y) = 0 时找水平线段，表达式无定义时找垂直线段。利用这些关键特征来匹配方程。
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