📚 Differential Equations for A-Level CCEA Maths | A-Level CCEA 数学：微分方程 考点精讲
Differential equations are a core part of the CCEA A-Level Mathematics specification, appearing in both the pure and applied strands. Mastering them requires fluency with first-order separable and linear types, second-order linear equations with constant coefficients, and the ability to model real-world situations. This revision guide breaks down every essential skill, from finding complementary functions and particular integrals to using initial conditions and avoiding common pitfalls.
微分方程是 CCEA A-Level 数学大纲的核心内容，出现在纯数学和应用数学部分。掌握这部分内容需要熟练处理一阶可分离和线性方程、二阶常系数线性方程，并能够对现实情境建立模型。本复习指南将逐一击破每个关键技能，从求补函数和特解到使用初始条件以及避开常见失分点。
1. What is a Differential Equation? | 什么是微分方程？
A differential equation is any equation that involves an unknown function and its derivatives. For example, dy/dx = 3x² is a simple first-order equation whose solution is found by direct integration. The order of a differential equation is the highest derivative that appears; A-Level work focuses on first-order (dy/dx) and second-order (d²y/dx²) equations.
微分方程是包含未知函数及其导数的方程。例如，dy/dx = 3x² 是一个简单的一阶方程，其解可通过直接积分求得。微分方程的阶数是其中出现的最高阶导数；A-Level 阶段重点研究一阶（dy/dx）和二阶（d²y/dx²）方程。
A general solution contains arbitrary constants (one constant for a first-order equation, two for a second-order equation). A particular solution is obtained when additional conditions, such as initial or boundary conditions, are used to determine these constants.
通解包含任意常数（一阶方程一个常数，二阶方程两个常数）。当用初始条件或边界条件等附加条件确定这些常数后，就得到特解。
2. First-Order Separable Equations | 一阶可分离变量方程
If a first-order equation can be written in the form dy/dx = g(x)h(y), it is separable. Rewrite it as (1/h(y)) dy = g(x) dx and integrate both sides. Always include one constant of integration on one side, and simplify to y = f(x) if possible.
若一阶方程可写成 dy/dx = g(x)h(y) 的形式，它就是可分离变量的。将其改写为 (1/h(y)) dy = g(x) dx，并对两边积分。始终在一边加一个积分常数，并尽可能化简为 y = f(x) 的形式。
Example: dy/dx = 2xy. Separate to get (1/y) dy = 2x dx, integrate to ln|y| = x² + C, then y = Ae^(x²) where A = ±e^C.
例如：dy/dx = 2xy。分离得 (1/y) dy = 2x dx，积分得 ln|y| = x² + C，然后 y = Aeˣ²，其中 A = ±e^C。
3. First-Order Linear Equations and Integrating Factors | 一阶线性方程与积分因子
A first-order linear equation has the standard form dy/dx + P(x)y = Q(x). The integrating factor (I.F.) is e^(∫P(x) dx). Multiply every term by the I.F., so the left-hand side becomes the exact derivative d/dx(y × I.F.). Then integrate both sides with respect to x and solve for y.
一阶线性方程标准形式为 dy/dx + P(x)y = Q(x)。积分因子 (I.F.) 为 e^(∫P(x) dx)。用积分因子乘每一项，使左边成为全导数 d/dx(y × I.F.)。然后两边对 x 积分并解出 y。
Example: dy/dx + 2y = e⁻ˣ. Here P(x)=2, so I.F. = e^(∫2 dx) = e²ˣ. Multiply through: e²ˣ dy/dx + 2e²ˣ y = eˣ. The left side is d/dx(y e²ˣ). Integrate: y e²ˣ = ∫eˣ dx = eˣ + C. Finally, y = e⁻ˣ + Ce⁻²ˣ.
例如：dy/dx + 2y = e⁻ˣ。这里 P(x)=2，I.F. = e^(∫2 dx) = e²ˣ。乘开：e²ˣ dy/dx + 2e²ˣ y = eˣ。左边是 d/dx(y e²ˣ)。积分得：y e²ˣ = ∫eˣ dx = eˣ + C。最后 y = e⁻ˣ + Ce⁻²ˣ。
4. Second-Order Linear Homogeneous Equations | 二阶线性齐次方程
A second-order linear differential equation with constant coefficients is homogeneous when the right-hand side is zero: a d²y/dx² + b dy/dx + c y = 0. To solve it, assume a solution of the form y = e^(λx), which leads to the auxiliary (or characteristic) equation: aλ² + bλ + c = 0.
二阶常系数线性微分方程当右边为零时是齐次的：a d²y/dx² + b dy/dx + c y = 0。求解时设解为 y = e^(λx)，从而得到辅助（特征）方程：aλ² + bλ + c = 0。
The roots λ₁ and λ₂ of this quadratic determine the form of the complementary function (C.F.). This function is the general solution of the homogeneous equation and contains two arbitrary constants.
该二次方程的根 λ₁ 和 λ₂ 决定了补函数 (C.F.) 的形式。这个函数是齐次方程的通解，包含两个任意常数。
5. Forms of the Complementary Function | 补函数的形式
The nature of the roots gives three distinct cases for the complementary function y_c:
根的性质使补函数 y_c 有三种不同情况：
| Roots of aλ²+bλ+c=0 | Complementary Function y_c |
| Real and distinct: λ₁ ≠ λ₂ | y_c = A e^(λ₁x) + B e^(λ₂x) |
| Real and equal: λ₁ = λ₂ = λ | y_c = (A + Bx) e^(λx) |
| Complex conjugate: λ = α ± iβ | y_c = e^(αx) (A cos βx + B sin βx) |
For complex roots, remember that the real part α governs exponential growth or decay, while the imaginary part β controls oscillation. This form is heavily tested in CCEA papers, especially with trigonometric forcing functions in the non-homogeneous case.
对于复根，记住实部 α 控制指数增长或衰减，而虚部 β 控制振荡。这种形式在 CCEA 考试中很常见，尤其是当非齐次项为三角函数时。
6. Non-Homogeneous Second-Order Equations | 非齐次二阶方程
A non-homogeneous equation has the form a d²y/dx² + b dy/dx + c y = f(x). Its general solution is y = y_c + y_p, where y_c is the complementary function (solution of the homogeneous equation) and y_p is a particular integral (any solution that satisfies the full equation).
非齐次方程形如 a d²y/dx² + b dy/dx + c y = f(x)。其通解为 y = y_c + y_p，其中 y_c 是补函数（齐次方程的解），y_p 是一个特解（满足整个方程的任意解）。
To find y_p, the method of undetermined coefficients is used: guess a form based on f(x), differentiate, substitute into the left-hand side, and equate coefficients to determine the unknown constants. CCEA expects you to handle polynomial, exponential, and trigonometric f(x).
为求 y_p，使用待定系数法：根据 f(x) 猜测一个形式，求导，代入左边，并比较系数来确定未知常数。CCEA 要求能处理多项式、指数和三角函数的 f(x)。
7. Guessing the Particular Integral | 特解的猜测规则
The trial function y_p must mirror f(x), but with an important modification: if any term in the trial function already appears in the complementary function, multiply the trial function by x (or x² if the root is repeated) to avoid duplication. This is the modification rule.
试探函数 y_p 必须与 f(x) 形式相同，但有一项重要修改：若试探函数中的任何项已在补函数中出现，则需乘以 x（若根为重根则乘以 x²）以避免重复。这就是修改规则。
Typical pairings: for f(x) = polynomial of degree n, try a general polynomial of the same degree; for f(x) = e^(kx), try y_p = Ae^(kx); for f(x) = sin ωx or cos ωx, try y_p = P cos ωx + Q sin ωx (even if only one trig function appears). For f(x) = e^(kx) cos ωx, try y_p = e^(kx)(P cos ωx + Q sin ωx). Always write the full trial expression and then find P, Q, etc.
典型组合：f(x) 为 n 次多项式时，尝试同次的一般多项式；f(x) = e^(kx) 时，尝试 y_p = Ae^(kx)；f(x) = sin ωx 或 cos ωx 时，尝试 y_p = P cos ωx + Q sin ωx（即使只出现一个三角函数也要设两项）。对于 f(x) = e^(kx) cos ωx，尝试 y_p = e^(kx)(P cos ωx + Q sin ωx)。务必写出完整的试探表达式再求 P, Q 等。
8. Initial and Boundary Conditions | 初始条件与边界条件
Once the general solution y = y_c + y_p is found, use given conditions to find the arbitrary constants. For a first-order equation, one condition (e.g., y(x₀)=y₀) is needed. For a second-order equation, two conditions are required; they can be initial conditions (y and dy/dx at the same x) or boundary conditions (y at two different x values).
找到通解 y = y_c + y_p 后，用给定条件求任意常数。一阶方程需要一个条件（如 y(x₀)=y₀）。二阶方程需要两个条件；它们可以是初始条件（同一点处的 y 和 dy/dx）或边界条件（两个不同 x 处的 y 值）。
Always differentiate the general solution first before substituting conditions, and solve the resulting simultaneous equations carefully. A common mistake is to forget to differentiate the particular integral when applying initial rates.
一定要先对通解求导，再代入条件，然后仔细解联立方程。常见的错误是在代入初始速率时忘记对特解求导。
9. Modelling with Differential Equations | 用微分方程建模
CCEA exam questions often embed differential equations within a real-world context, such as population growth, cooling, mixing problems, or mechanical oscillations. The process involves interpreting the description to form an equation, solving it, and then interpreting the solution back in context, including units and long-term behaviour.
CCEA 考试常将微分方程嵌在实际情境中，如人口增长、冷却、混合问题或机械振动。过程包括解读描述以建立方程、求解方程、再结合上下文解释解的意义，包括单位和长期行为。
Key phrases: ‘rate of change’ means derivative; ‘proportional to’ gives a constant of proportionality k. For Newton’s law of cooling, dθ/dt = -k(θ – θ₀). For exponential growth/decay, dN/dt = kN. Always identify variables and their initial values clearly before starting.
关键短语：“变化率”意味着导数；“正比于”给出比例常数 k。对于牛顿冷却定律，dθ/dt = -k(θ – θ₀)。对于指数增长/衰减，dN/dt = kN。在开始前务必明确变量及其初始值。
10. Full Solution Walkthrough | 完整解题示例
Let’s solve: d²y/dx² – 3 dy/dx + 2y = 4eˣ, given y(0)=1, y'(0)=0.
求解：d²y/dx² – 3 dy/dx + 2y = 4eˣ，已知 y(0)=1, y'(0)=0。
Step 1: Auxiliary equation λ² – 3λ + 2 = 0 → (λ-1)(λ-2)=0 → λ=1,2. So y_c = A eˣ + B e²ˣ.
步骤1：辅助方程 λ² – 3λ + 2 = 0 → (λ-1)(λ-2)=0 → λ=1,2。因此 y_c = A eˣ + B e²ˣ。
Step 2: f(x)=4eˣ, which matches part of y_c (eˣ). Multiply the standard trial y_p = C eˣ by x to get y_p = Cx eˣ. Differentiate: y_p’ = C eˣ + Cx eˣ, y_p” = 2C eˣ + Cx eˣ. Substitute into left side: (2C eˣ + Cx eˣ) -3(C eˣ + Cx eˣ) +2(Cx eˣ) = 4eˣ. Simplify: 2C eˣ – 3C eˣ = -C eˣ, so -C eˣ = 4eˣ ⇒ C = -4. Hence y_p = -4x eˣ.
步骤2：f(x)=4eˣ，与 y_c 中某项（eˣ）重合。将标准试探 y_p = C eˣ 乘以 x，得 y_p = Cx eˣ。求导：y_p’ = C eˣ + Cx eˣ，y_p” = 2C eˣ + Cx eˣ。代入左边：(2C eˣ + Cx eˣ) – 3(C eˣ + Cx eˣ) + 2(Cx eˣ) = 4eˣ。化简：2C eˣ – 3C eˣ = -C eˣ，因此 -C eˣ = 4eˣ ⇒ C = -4。于是 y_p = -4x eˣ。
Step 3: General solution y = A eˣ + B e²ˣ – 4x eˣ. Use conditions: y(0)= A + B = 1. Differentiate: y’ = A eˣ + 2B e²ˣ – 4 eˣ – 4x eˣ. y'(0)= A + 2B -4 = 0. Solve: B = 3, A = -2. Particular solution: y = -2 eˣ + 3 e²ˣ – 4x eˣ.
步骤3：通解 y = A eˣ + B e²ˣ – 4x eˣ。利用条件：y(0)= A + B = 1。求导：y’ = A eˣ + 2B e²ˣ – 4 eˣ – 4x eˣ。y'(0)= A + 2B -4 = 0。解得：B = 3, A = -2。特解：y = -2 eˣ + 3 e²ˣ – 4x eˣ。
11. Common Mistakes and How to Avoid Them | 常见错误与避坑指南
Mistake 1: Forgetting the constant of integration in separable equations. Always add +C immediately after integrating; don’t wait until the end.
错误1：在可分离方程中忘记积分常数。积分后立刻加上 +C，不要拖到最后。
Mistake 2: Not checking whether the trial y_p overlaps with y_c. If f(x) has the same form as any term in y_c, multiply by the independent variable (x). Repeated roots require x².
错误2：未检查试探 y_p 是否与 y_c 有重叠。如果 f(x) 与 y_c 中某项形式相同，乘以自变量（x）。重根需要乘以 x²。
Mistake 3: Using the wrong trial for trigonometric f(x). Always include both sine and cosine, even if the non-homogeneous term contains only one of them.
错误3：在三角函数 f(x) 时用错试探形式。即使非齐次项只含 sin 或 cos，也要同时包含正弦和余弦。
Mistake 4: Algebraic slips when substituting y_p into the differential equation. Write the left-hand side as a clear collection of like terms.
错误4：将 y_p 代入微分方程时出现代数失误。将左边清晰地按同类项整理。
12. Exam Tips and Final Review | 考试技巧与总结回顾
CCEA A-Level papers often include a multi-part question: (a) solve the homogeneous equation, (b) find a particular integral, (c) apply conditions. Follow the steps systematically. Show the auxiliary equation, the roots, and the form of y_c explicitly to earn method marks.
CCEA A-Level 试卷常包含多部分问题：(a) 求解齐次方程，(b) 求特解，(c) 应用条件。按步骤系统作答。明确写出辅助方程、根和 y_c 的形式以获取过程分。
Memorise the three complementary function forms and the table of trial particular integrals. Practice recognition: if f(x) is a product of polynomial and exponential, adjust accordingly. Always check your final particular solution by substituting back into the original equation if time permits.
熟记三种补函数形式和特解试探表。练习识别：若 f(x) 是多项式和指数的乘积，相应调整。如果时间允许，总要将最终特解代回原方程检验。
For modelling questions, define variables, state assumptions, and interpret your final mathematical result in words. Use correct units and comment on the validity of the model, especially for large t.
对于建模题，定义变量，陈述假设，并用文字解释最终数学结果。使用正确的单位，并对模型的合理性进行评论，尤其是当 t 很大时。
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