📚 A-Level Maths: Differential Equations Exam Tips | A-Level 数学：微分方程 考点精讲
Differential equations are the language of change. In A-Level Mathematics, they bridge algebra, calculus and real-world modelling, appearing regularly in both Pure and Applied papers. This article unpacks the key techniques you must master, from separating variables to solving second-order non-homogeneous equations with confidence.
微分方程是描述变化的语言。在 A-Level 数学中，它连接了代数、微积分与现实建模，是纯数和应用卷中的常客。本文帮你梳理从分离变量到二阶非齐次方程的全部核心技巧，助你自信应考。
1. What is a Differential Equation? | 什么是微分方程？
A differential equation (DE) relates a function to its derivatives. For A-Level, the independent variable is usually x and the dependent variable is y, so you will see expressions like dy/dx, d²y/dx² or sometimes dy/dt for rates of change with respect to time t.
微分方程是未知函数与其导数之间的关系式。A-Level 中自变量多为 x，因变量为 y，因此常见 dy/dx、d²y/dx²，有时用 dy/dt 表示对时间的变化率。
The order of a DE is the highest derivative present. A first-order DE involves only dy/dx; a second-order DE involves d²y/dx². Understanding the order tells you which solving strategy to use.
方程的阶由出现的最高阶导数决定。一阶方程只含 dy/dx，二阶方程含有 d²y/dx²。确定阶数能帮你锁定正确的求解策略。
2. Types of First-Order DEs | 一阶微分方程的类型
The four main types you will meet at A-Level are: separable, linear, homogeneous (in the sense of y/x substitution) and exact (depending on the specification). Recognising the type quickly is half the battle, because each has a structured solution method.
A-Level 阶段主要涉及四种一阶方程：可分离变量型、线性型、齐次型（指可用 y/x 替换）以及恰当方程（视考纲而定）。快速识别类型等于成功了一半，因为每种类型都有固定的解法流程。
Always rewrite the DE in the form dy/dx = f(x, y) and inspect the structure. If you can factorise f(x, y) as g(x)h(y), it is separable. If it matches dy/dx + P(x)y = Q(x), it is linear. If f(x, y) can be expressed as a function of y/x only, it is homogeneous.
答题时先把方程写成 dy/dx = f(x, y) 的形式并观察结构。若能分解为 g(x)h(y)，就是可分离型；若能整理成 dy/dx + P(x)y = Q(x)，就是线性；若 f(x, y) 可写成仅含 y/x 的函数，就是齐次型。
3. Separable Equations | 可分离变量方程
For a separable DE dy/dx = g(x)h(y), bring all y terms to one side and all x terms to the other: (1/h(y)) dy = g(x) dx. Then integrate both sides. Never forget the constant of integration – it is usually added immediately to the x-side as +C.
对于可分离方程 dy/dx = g(x)h(y)，将所有含 y 的项移到一边、含 x 的项移到另一边：(1/h(y)) dy = g(x) dx，然后两边积分。千万不要忘记积分常数——通常直接加在 x 侧，写作 +C。
∫ 1/h(y) dy = ∫ g(x) dx + C
If h(y) contains a factor like (y – 3), the integral produces ln|y – 3|. A-Level marks expect the modulus signs or explicit handling of domain. You may also be asked to give the solution in explicit form y = f(x) if possible.
若 h(y) 含有 (y – 3) 之类的因子，积分会出现 ln|y – 3|。A-Level 评分标准要求写出绝对值记号或明确说明定义域。如果可能，还需要将解写成显函数形式 y = f(x)。
Common example: dy/dx = ky, giving y = Aeᵏˣ, which is exponential growth or decay. Exam questions often ask for the particular value of A using an initial condition, so substitute x = 0, y = y₀ to find A.
常见例子：dy/dx = ky，解得 y = Aeᵏˣ，代表指数增长或衰减。考题常要求代入初始条件求特解，例如设 x = 0, y = y₀ 来确定 A。
4. Linear Equations and Integrating Factor | 线性方程与积分因子
A first-order linear DE has the standard form dy/dx + P(x)y = Q(x). The integrating factor (IF) is μ(x) = e^(∫ P(x) dx). Multiply the entire equation by μ(x); the left-hand side becomes d/dx (μ(x)y), so you can integrate both sides directly.
一阶线性方程的标准形式为 dy/dx + P(x)y = Q(x)。积分因子 μ(x) = e^(∫ P(x) dx)。将整个方程乘以 μ(x)，左边就变成 d/dx (μ(x)y)，因此可以直接对两边积分。
μ(x) = e∫P(x)dx, d/dx [μ(x)y] = μ(x)Q(x) ⇒ y = (1/μ) ∫ μ Q dx
Choose the simplest form of μ without adding ‘+C’ to the exponent, as any version works. After integration, include +C and then divide by μ to obtain the general solution. Always check if the question requires the answer in a specific form, like y = (x² + C)/x.
计算 μ 时选择最简单的形式，指数部分无需加 +C，因为任何形式都有效。积分后加上常数 C，再除以 μ 即得通解。务必检查题目是否要求将答案整理成特定形式，如 y = (x² + C)/x。
In exam settings, a common pitfall is forgetting to multiply the right-hand side Q(x) by μ(x) before integrating. Another is algebraic slip when simplifying the final expression.
考试中常见的错误是忘记在积分前将右边 Q(x) 也乘上 μ(x)，或者在化简最终表达式时出现代数错误。
5. Homogeneous Equations (y/x substitution) | 齐次方程（y/x 替换）
A first-order DE is homogeneous if it can be written as dy/dx = F(y/x). Use the substitution v = y/x, so that y = v x and dy/dx = v + x dv/dx. The original equation turns into a separable equation in v and x.
若一阶方程可写成 dy/dx = F(y/x) 的形式，就是齐次方程。使用代换 v = y/x，则 y = v x，且 dy/dx = v + x dv/dx。原方程就转化为关于 v 和 x 的可分离方程。
v + x dv/dx = F(v) ⇒ ∫ 1/(F(v)–v) dv = ∫ 1/x dx
After solving for v, replace back v = y/x to get the solution in terms of x and y. Sometimes the final answer is left implicitly; this is acceptable unless an explicit form is requested.
解出 v 后，代入 v = y/x 还原成 x 与 y 的关系。最终的解有时可保留隐式形式，除非题目明确要求显式表达。
Remember to check whether the DE is truly of the form dy/dx = F(y/x). Expressions like (x + y)/(x – y) clearly qualify, while (x² + y) do not. Misidentifying the type wastes valuable time.
注意验证方程是否确实为 dy/dx = F(y/x) 的形式。像 (x + y)/(x – y) 显然符合，而 (x² + y) 则不然。误判类型会浪费宝贵的考试时间。
6. Exact Equations (if required) | 恰当方程（如考纲要求）
Some specifications, like Edexcel’s Further Pure, include exact differential equations. An equation M(x,y)dx + N(x,y)dy = 0 is exact if ∂M/∂y = ∂N/∂x. The solution is found by partial integration: find a function ψ(x,y) such that ∂ψ/∂x = M and ∂ψ/∂y = N, then set ψ = C.
某些考试局（如 Edexcel 的进阶纯数）包含恰当微分方程。方程 M(x,y)dx + N(x,y)dy = 0 若满足 ∂M/∂y = ∂N/∂x 就是恰当方程。解法是通过偏积分构造函数 ψ(x,y) 使得 ∂ψ/∂x = M、∂ψ/∂y = N，然后设 ψ = C。
The test for exactness is quick: compute partial derivatives. If true, integrate M with respect to x, adding an arbitrary function of y, then differentiate with respect to y and compare with N to determine that function.
恰当性检验很快捷：计算偏导数是否相等。若成立，对 x 积分 M，添加关于 y 的任意函数，再对 y 求导，与 N 比较以确定该函数。
Exact equations often appear in contexts like orthogonal trajectories or conservative fields. Even if not on your core paper, understanding the concept deepens your overall DE intuition.
恰当方程常出现在正交轨线或保守场等背景中。即使你考的核心卷不涉及，理解这一概念也能加深对微分方程的整体直觉。
7. Second-Order Linear Homogeneous DEs | 二阶常系数齐次线性方程
These have the form a d²y/dx² + b dy/dx + c y = 0, where a, b, c are constants. The method is to assume a solution of the form y = e^(λx), leading to the auxiliary equation a λ² + b λ + c = 0. The nature of roots dictates the general solution.
这类方程形式为 a d²y/dx² + b dy/dx + c y = 0，其中 a, b, c 是常数。解法是假设解为 y = e^(λx)，代入得到辅助方程 a λ² + b λ + c = 0。根的性质决定通解的形式。
a λ² + b λ + c = 0
If the auxiliary equation has two distinct real roots α and β, the general solution is y = A e^(αx) + B e^(βx). For a repeated root λ (discriminant zero), the solution is y = (A + Bx) e^(λx). For complex conjugate roots p ± iq, the solution is y = e^(px) (A cos(qx) + B sin(qx)).
若辅助方程有两个相异实根 α 与 β，通解为 y = A e^(αx) + B e^(βx)。若为重根 λ（判别式为零），解为 y = (A + Bx) e^(λx)。若为共轭复根 p ± iq，解为 y = e^(px) (A cos(qx) + B sin(qx))。
Always write the solution with two arbitrary constants A and B. The values are later determined by initial conditions or boundary conditions, such as y(0) = 1, y'(0) = 0.
解式中务必保留两个任意常数 A 与 B。它们的取值将在后续由初始条件或边界条件确定，例如 y(0) = 1, y'(0) = 0。
8. Non-Homogeneous Second-Order DEs | 二阶非齐次方程
The equation is a d²y/dx² + b dy/dx + c y = f(x). The general solution is y = y_c + y_p, where y_c is the complementary function (solve the homogeneous part) and y_p is a particular integral (PI) that depends on f(x).
方程为 a d²y/dx² + b dy/dx + c y = f(x)。通解为 y = y_c + y_p，其中 y_c 是补函数（对应齐次部分的解），y_p 是依赖于 f(x) 的特解。
The method of undetermined coefficients is used: guess a form for y_p based on f(x). For a polynomial f(x), try a polynomial of the same degree. For e^(kx), try C e^(kx). For sin ωx or cos ωx, try C cos ωx + D sin ωx. If the guess overlaps with the complementary function, multiply by x (or x²).
采用待定系数法：根据 f(x) 猜测 y_p 的形式。若 f(x) 为多项式，尝试同次多项式；若为 e^(kx)，尝试 C e^(kx)；若为 sin ωx 或 cos ωx，尝试 C cos ωx + D sin ωx。如果试探形式与补函数重叠，则乘以 x（或 x²）。
If y_c contains eᵏˣ, and f(x) = eᵏˣ, then try y_p = C x eᵏˣ.
After substituting y_p and its derivatives into the LHS, equate coefficients to find the constants. Finally, combine y_c and y_p, and apply given conditions to find A and B.
将 y_p 及其导数代入方程左边，比较系数求出待定常数。最后将 y_c 与 y_p 合并，并根据已知条件求出 A 和 B。
9. Initial and Boundary Conditions | 初始与边界条件
Initial conditions typically give the values of y and/or dy/dx at a single point, often x = 0. Boundary conditions give information at two different x-values. Both are used to pin down arbitrary constants.
初始条件通常给出 y 和/或 dy/dx 在某一点（常为 x = 0）的值。边界条件则给出两个不同 x 处的信息。两者都用来确定任意常数。
Work systematically: first write the general solution with constants A and B. Then differentiate if necessary. Substitute the given x and y values to form simultaneous equations for A and B. Solve them accurately; arithmetic errors here are costly but avoidable.
按部就班：先写出含有 A、B 的通解，如需再求导。代入已知 x、y 值，得到关于 A、B 的方程组。仔细求解，这里的算术错误虽常见但完全可以避免。
For second-order DEs, two conditions are needed. A common exam task is to verify that a given function satisfies the DE and the conditions; always show substitution clearly.
对于二阶方程，需要两个条件。典型的考题是验证某给定函数是否满足方程及条件，务必清晰展示代入过程。
10. Modelling with Differential Equations | 微分方程建模
Many real-world scenarios are modelled by DEs: population growth (dP/dt = kP), cooling (dθ/dt = -k(θ – θ₀)), mixing problems, and simple harmonic motion (d²x/dt² = -ω²x). Translating a word problem into a DE is a critical skill.
许多现实情境可用微分方程建模：人口增长 (dP/dt = kP)、冷却定律 (dθ/dt = -k(θ – θ₀))、混合问题以及简谐运动 (d²x/dt² = -ω²x)。把文字描述转化为微分方程是核心技能。
Identify the rate of change, the proportionality constant, and any external inputs. Often you need to express ‘the rate of increase of P is proportional to P’ as dP/dt = kP. If there is a carrying capacity, the logistic model appears, but A-Level typically stays with simpler linear or separable models.
找出变化率、比例常数及外部干扰项。常见表述“P 的增长率与 P 成正比”对应 dP/dt = kP。若涉及环境容纳量，会出现逻辑斯蒂模型，不过 A-Level 通常只要求较简单的线性或可分离模型。
After solving, interpret the solution in context. Give the long-term behaviour or the time when a certain value is reached. Always use correct units and check if the answer is physically reasonable.
解出方程后，要在实际情境中解读解的意义，如长期趋势或达到某值的时间。始终使用正确单位，并核查答案在现实中是否合理。
11. Common Mistakes and How to Avoid Them | 常见错误与避坑指南
Mistake 1: Dropping the constant of integration. Always write +C immediately upon integrating, and only then apply conditions. Mistake 2: Incorrectly identifying the type of DE, leading to a doomed method. Pause and inspect the form before starting.
错误一：漏掉积分常数。一旦积分就立刻写下 +C，之后再代入条件。错误二：误判方程类型，导致解题方法完全错误。动笔前先暂停，仔细辨认方程形式。
Mistake 3: In linear DEs, forgetting that the integrating factor must multiply the entire Q(x) before integration. Mistake 4: In second-order non-homogeneous, failing to modify the trial function when overlap with y_c occurs.
错误三：线性方程中，忘记在积分前将积分因子乘到整项 Q(x) 上。错误四：二阶非齐次方程中，当试探解与补函数重叠时未做修正。
Mistake 5: In modelling, misinterpreting proportional relationships. ‘Rate of decrease proportional to square root’ means dX/dt = -k √X, not dX/dt = -k X². Read the wording carefully.
错误五：建模时误解比例关系。“减少速率与平方根成正比”意味着 dX/dt = -k √X，而不是 -k X²。务必仔细审题。
| Error | Quick fix |
|---|---|
| Missing modulus in ln|y| | Always include abs values initially |
| Incorrect auxiliary equation | Write λ² + (b/a)λ + (c/a)=0, double-check signs |
| PI guess wrong form | Check overlap with y_c; multiply by x if needed |
12. Exam Strategy and Revision Tips | 考试策略与复习建议
Start by reading the question fully: is it set in a modelling context? Are initial conditions given explicitly? Underline key instructions like ‘Show that…’, ‘Find the particular solution’, or ‘Give answer in the form y = …’.
先通读题目：是否包含建模背景？初始条件是否明确给出？圈出关键指令如“证明……”、“求特解”或“以 y = … 的形式给出答案”。
When verifying a solution, substitute it directly into the DE and simplify – don’t re-derive it. For method marks, even if you make a numerical slip, a clear structure will still earn credit.
当要验证某个解时，直接代入方程并化简，不必重新推导。即使出现计算失误，清晰的结构也能为你争取到方法分。
Practise past paper questions in timed conditions, focusing on the transition between DE types. Create flash cards for the forms of trial PIs and the general solutions of second-order homogeneous equations. Speed and accuracy with integration and differentiation are assumed, so keep those skills sharp.
在计时条件下练习历年真题，重点训练不同方程类型之间的切换。制作记忆卡片，记牢试探特解的形式和二阶齐次方程的通解公式。考试假定你的微积分运算又快又准，因此要保持这些基本技能不生疏。
If you get stuck, write down the relevant general solution or the separation step – often this alone earns a mark. Above all, a calm, methodical approach transforms differential equations from intimidating to manageable.
如果卡住了，至少写出相关的通解形式或变量分离步骤，这本身常常就能得分。最重要的是，冷静、有条不紊的方法能让你把微分方程从令人畏惧变得应对自如。
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