Differential Equations for A-Level WJEC Maths | A-Level WJEC 数学：微分方程考点精讲

📚 Differential Equations for A-Level WJEC Maths | A-Level WJEC 数学：微分方程考点精讲

Differential equations are a cornerstone of A-Level Mathematics, bridging the gap between pure calculus and real-world applications. In the WJEC specification, you are expected to master techniques for solving first-order and second-order differential equations, understand how to formulate them from given contexts, and apply these solutions in practical models. This guide breaks down every essential topic, from separation of variables to integrating factors and second-order homogeneous equations, complete with worked examples and exam strategies.

微分方程是 A-Level 数学的核心内容，它连接了纯微积分和实际应用。在 WJEC 考纲中，你需要掌握求解一阶和二阶微分方程的方法，理解如何根据给定情境建立方程，并能在实际模型中应用这些解。本篇指南将逐一解析每个必考主题，从分离变量法到积分因子，再到二阶齐次方程，并辅以例题和应试策略。

1. Introduction to Differential Equations | 微分方程简介

A differential equation involves an unknown function and its derivatives. The order is the highest derivative present. For example, dy/dx + 2y = 0 is first order, while d²y/dx² + 3dy/dx + 2y = 0 is second order. The general solution contains arbitrary constants, while a particular solution satisfies given initial or boundary conditions.

微分方程包含未知函数及其导数。阶数指方程中最高阶导数的阶。例如，dy/dx + 2y = 0 是一阶方程，而 d²y/dx² + 3dy/dx + 2y = 0 是二阶方程。通解包含任意常数，而特解则满足给定的初始条件或边界条件。

All the techniques you will learn rely on integration. The key is to rewrite the equation so that both sides can be integrated directly, or to use a multiplier that makes the left-hand side an exact derivative.

你将要学习的所有技巧都依赖于积分。关键是将方程改写为两边可直接积分的形式，或者使用一个乘子将左边化成全导数。

2. Forming Differential Equations | 建立微分方程

WJEC exam questions often ask you to construct a differential equation from a written description. The phrase ‘rate of change of … is proportional to …’ translates to dy/dt ∝ y, giving dy/dt = ky where k is a constant. If it is inversely proportional, use dy/dt = k/y or similar.

WJEC 考题经常要求根据文字描述建立微分方程。“…的变化率与…成正比”对应 dy/dt ∝ y，得 dy/dt = ky，其中 k 为常数。若为反比，则使用 dy/dt = k/y 或类似形式。

Another common context is a mixing problem: ‘The rate at which salt leaves the tank is proportional to the amount of salt present at time t,’ leading to dA/dt = -kA. Always define your variables clearly and state the constant of proportionality.

另一个常见背景是混合问题：“盐离开水箱的速率与 t 时刻水箱中的盐量成正比”，得到 dA/dt = -kA。务必清晰定义变量，并写出比例常数。

3. Separation of Variables | 分离变量法

This method works for equations of the form dy/dx = f(x)g(y). You collect all y-terms on the left with dy and all x-terms on the right with dx:

(1/g(y)) dy = f(x) dx

这种方法适用于形如 dy/dx = f(x)g(y) 的方程。将所有含 y 的项与 dy 移到左边，含 x 的项与 dx 移到右边，得：(1/g(y)) dy = f(x) dx

Then integrate both sides. Don’t forget the constant of integration on one side. For example, if dy/dx = xy, we separate to (1/y) dy = x dx, integrate to ln|y| = ½x² + C, and solve for y = Ae^(½x²) where A = ±e^C.

然后两边积分。切勿忘记在一侧加上积分常数。例如 dy/dx = xy，分离得 (1/y) dy = x dx，积分得 ln|y| = ½x² + C，解出 y = Ae^(½x²)，其中 A = ±e^C。

Always check whether y can be zero; the separable method assumes g(y) ≠ 0. Lost solutions need to be considered, but in WJEC problems the general solution usually covers all cases once you allow A=0.

务必检查 y 是否可能为零；分离变量法假设 g(y) ≠ 0。需要考虑丢失的解，但在 WJEC 问题中，一旦允许 A=0，通解通常涵盖所有情形。

4. First Order Linear Differential Equations | 一阶线性微分方程

The standard form is dy/dx + P(x)y = Q(x). Here P(x) and Q(x) are functions of x only. The key is to find an integrating factor that turns the left-hand side into the derivative of a product.

标准形式为 dy/dx + P(x)y = Q(x)。其中 P(x) 和 Q(x) 仅为 x 的函数。关键是要找到一个积分因子，使左边变为两个函数乘积的导数。

This form arises in many modelling contexts, such as Newton’s law of cooling or electrical circuits. The method that follows is the standard WJEC route for solving these equations exactly.

这种形式出现在许多建模场景中，如牛顿冷却定律或电路问题。接下来的方法是 WJEC 求解此类方程的标准途径。

5. Integrating Factor Method | 积分因子法

The integrating factor I(x) is given by:

I(x) = e^(∫ P(x) dx)

积分因子 I(x) 计算公式为：I(x) = e^(∫ P(x) dx)

Multiply the entire differential equation by I(x). The left-hand side becomes d/dx[ I(x)·y ]. You then integrate both sides with respect to x: I(x)·y = ∫ I(x)Q(x) dx. Finally, solve for y by dividing through by I(x).

将整个微分方程乘以 I(x)。左边即变为 d/dx[ I(x)·y ]。然后两边对 x 积分：I(x)·y = ∫ I(x)Q(x) dx。最后两边除以 I(x) 解出 y。

Example: dy/dx + 2y = eˣ. Here P(x)=2, I(x)=e^(∫2 dx)=e²ˣ. Multiply: e²ˣ dy/dx + 2e²ˣ y = e³ˣ, so d/dx(e²ˣ y) = e³ˣ. Integrate: e²ˣ y = (1/3)e³ˣ + C. So y = (1/3)eˣ + Ce⁻²ˣ.

示例：dy/dx + 2y = eˣ。这里 P(x)=2，I(x)=e^(∫2 dx)=e²ˣ。相乘得 e²ˣ dy/dx + 2e²ˣ y = e³ˣ，即 d/dx(e²ˣ y) = e³ˣ。积分：e²ˣ y = (1/3)e³ˣ + C。故 y = (1/3)eˣ + Ce⁻²ˣ。

Always simplify the integrating factor by ignoring the constant of integration in the exponent, as it cancels out when multiplying through.

计算积分因子时，指数中的积分常数可以忽略，因为在相乘过程中会被约去。

6. Solving Second Order Homogeneous DEs with Constant Coefficients | 解二阶常系数齐次微分方程

These take the form a d²y/dx² + b dy/dx + c y = 0, where a, b, c are constants. Solve the auxiliary equation:

a m² + b m + c = 0

其形式为 a d²y/dx² + b dy/dx + c y = 0，其中 a、b、c 为常数。求解辅助方程：a m² + b m + c = 0

The form of the general solution depends on the discriminant Δ = b² – 4ac:

Two distinct real roots m₁, m₂: y = A e^(m₁ x) + B e^(m₂ x)
Repeated real root m: y = (A + B x) e^(m x)
Complex conjugate roots p ± iq: y = e^(p x) (A cos qx + B sin qx)

通解的形式取决于判别式 Δ = b² – 4ac：

两个不等实根 m₁、m₂： y = A e^(m₁ x) + B e^(m₂ x)
重根 m： y = (A + B x) e^(m x)
共轭复根 p ± iq： y = e^(p x) (A cos qx + B sin qx)

For WJEC, the complex case is often tested. Memorise the forms and practise recognising the correct one quickly. In the repeated root case, the extra x factor is crucial.

在 WJEC 中，共轭复根的情况常被考查。记住各形式并练习快速识别正确形式。重根情形中，多余的因子 x 至关重要。

7. Particular Solutions and Initial/Boundary Conditions | 特解与初始/边界条件

Given initial conditions (e.g. y = y₀ when x = 0), you can find the exact values of the arbitrary constants. For first order equations, one condition is enough; for second order, two conditions are needed, such as y(0) = 2 and y'(0) = 1.

给定初始条件（例如当 x = 0 时 y = y₀），可求出任意常数的具体值。一阶方程只需一个条件；二阶方程则需要两个条件，例如 y(0) = 2 和 y'(0) = 1。

Substitute the conditions into the general solution or its derivative, then solve the resulting simultaneous equations. Always present your final particular solution in its simplest form.

将条件代入通解或其导数，然后求解所得的方程组。始终以最简形式给出最终的特解。

8. Modelling with Differential Equations: Exponential Growth and Decay | 微分方程建模：指数增长与衰减

Exponential growth/decay is modelled by dy/dt = k y, where k > 0 for growth and k < 0 for decay. The general solution is y = A e^(k t). The constant A represents the initial value y(0).

指数增长/衰减的模型为 dy/dt = k y，其中 k > 0 为增长，k < 0 为衰减。通解为 y = A e^(k t)。常数 A 代表初始值 y(0)。

In word problems, you might be given a doubling time or half-life. For growth: if y doubles in T years, then 2 = e^(k T) ⇒ k = (ln 2)/T. For decay: half-life T₁/₂ gives k = -(ln 2) / T₁/₂.

在文字题中，可能会给出倍增时间或半衰期。对于增长：若 y 在 T 年内翻倍，则 2 = e^(k T) ⇒ k = (ln 2)/T。对于衰减：半衰期 T₁/₂ 给出 k = -(ln 2) / T₁/₂。

WJEC questions frequently ask you to find the time when a quantity reaches a certain value, or to express the constant k in exact logarithmic form.

WJEC 题目经常要求计算某量达到特定值所需的时间，或用精确的对数形式表示常数 k。

9. Contextual Problems: Newton’s Law of Cooling and Mixing | 应用题：牛顿冷却定律与混合问题

Newton’s Law of Cooling states: dT/dt ∝ (T – T_env) or dT/dt = -k(T – T_env), where T_env is the constant ambient temperature. This is a first order linear DE solved by separation or integrating factor.

牛顿冷却定律为：dT/dt ∝ (T – T_环境) 或 dT/dt = -k(T – T_环境)，其中 T_环境为恒定的室温。这是一阶线性微分方程，可用分离变量法或积分因子法求解。

Mixing problems often involve a tank with salt solution: A(t) is the amount of salt at time t. If water with concentration c kg/L flows in at rate r L/min, and the well-mixed solution flows out at the same rate, then dA/dt = c r – (r/V₀) A, assuming constant volume V₀.

混合问题常涉及含盐溶液的容器：A(t) 为 t 时刻的盐量。若浓度为 c kg/L 的水以速率 r L/min 流入，且充分混合的溶液以相同速率流出，则 dA/dt = c r – (r/V₀) A，假设容积 V₀ 恒定。

Set up the equation carefully, identify P and Q, then use an integrating factor. The general solution gives the long-term equilibrium value, which can also be found by setting dA/dt = 0.

需仔细建立方程，确定 P 和 Q，然后使用积分因子。通解可给出长期平衡值，该值也可通过令 dA/dt = 0 得到。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Mistake: Forgetting to include the constant of integration immediately after integrating. It must be added on one side before any algebraic manipulation. Add it as soon as you integrate.

常见错误：积分后忘记立即加入积分常数。必须在进行任何代数变形之前在一侧加上常数。积分后立即加上。

Mistake: Losing solutions when dividing by a function that could be zero, such as y in dy/dx = xy. Always state the assumption y ≠ 0 and check if y = 0 is a solution.

常见错误：除以可能为零的函数（如 dy/dx = xy 中的 y）时丢失解。务必声明假设 y ≠ 0，并检查 y = 0 是否为解。

Mistake: Using incorrect form for complex roots in second order DEs. Remember that the roots p ± iq lead to e^(p x) (A cos qx + B sin qx), not A e^(p x) cos qx + B e^(p x) sin qx which is the same but some write it incorrectly with extra constants.

常见错误：二阶微分方程中复根的形式写错。记住，根 p ± iq 得到 e^(p x) (A cos qx + B sin qx)，而不是 A e^(p x) cos qx + B e^(p x) sin qx（实际相同，但有人会错误地把常数位置弄混淆）。

Exam tip: Always show clear steps: write the standard form, find the integrating factor or auxiliary equation, state the general solution form, substitute conditions, and simplify. State final answers clearly boxed or underlined.

应试技巧：始终写出清晰步骤：写出标准形式，求出积分因子或辅助方程，写出通解形式，代入条件，化简。最终答案应方框或下划线明确标示。

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Differential Equations for A-Level WJEC Maths | A-Level WJEC 数学：微分方程 考点精讲