📚 Differential Equations Key Points | 微分方程 考点精讲
Differential equations form a crucial part of the IB and AQA A-Level Mathematics curricula, bridging pure calculus with real-world modelling. This article walks through the core techniques examined: separable equations, linear first-order equations with integrating factor, homogeneous equations, second-order linear equations with constant coefficients, and key applications. Whether you are preparing for Paper 3 or a synoptic exam, mastering these methods will give you a solid foundation.
微分方程是 IB 与 AQA 数学课程中连接纯微积分与现实建模的关键部分。本文将梳理考试涉及的核心方法：可分离变量方程、带积分因子的一阶线性方程、齐次方程、二阶常系数线性方程以及重要应用。无论你是在备考 IB 试卷三还是综合性考试，掌握这些方法都将奠定扎实的基础。
1. What is a Differential Equation? | 什么是微分方程？
A differential equation (DE) is an equation that involves an unknown function and its derivatives. It describes how a quantity changes, rather than just its static value. For example, dy/dx = 3y states that the rate of change of y is proportional to y itself.
微分方程是包含未知函数及其导数的方程。它描述的是某个量的变化规律，而不仅仅是该量的静态值。例如，dy/dx = 3y 表明 y 的变化率与 y 本身成正比。
The order of a DE is the highest derivative present. A first-order DE contains only first derivatives, while a second-order DE includes terms like y″ or d²y/dx². In IB and AQA exams, you will work with both orders and must be able to classify equations quickly.
微分方程的阶由其中出现的最高阶导数决定。一阶微分方程只包含一阶导数，而二阶方程则涉及 y″ 或 d²y/dx²。在 IB 和 AQA 考试中，你需要处理这两种阶，并能迅速对方程进行分类。
2. First-Order Separable Equations | 一阶可分离变量方程
A separable equation can be written in the form dy/dx = f(x)g(y), where the right-hand side is a product of a function of x alone and a function of y alone. The strategy is to treat dy/dx as a fraction and rearrange so that all terms involving y are on one side and all terms involving x are on the other.
可分离变量方程可写成 dy/dx = f(x)g(y) 的形式，即右侧是仅含 x 的函数与仅含 y 的函数的乘积。解题思路是将 dy/dx 当作分式处理，重新排列，使所有含 y 的项在一边、所有含 x 的项在另一边。
This gives ∫ (1/g(y)) dy = ∫ f(x) dx. After integrating both sides, you obtain an implicit or explicit general solution. Never forget to include the constant of integration; the general solution must contain one arbitrary constant for a first-order equation.
由此得到 ∫ (1/g(y)) dy = ∫ f(x) dx。两边积分后，即可得到隐式或显式的通解。切记要加上积分常数；对于一阶方程，通解必须包含一个任意常数。
3. Solving Separable Equations Step-by-Step | 可分离变量方程的逐步求解
Begin by rewriting the derivative dy/dx as an explicit fraction. Separate the variables, ensuring that no y terms remain on the dx side. Then integrate each side with respect to its own variable.
首先将导数 dy/dx 明确表示为分式。进行变量分离，确保 dx 一侧不再含有 y 的项。然后分别对各自变量进行积分。
If the integrations produce logarithms, remember that ln|y| = … leads to |y| = e^(…), and you can drop the absolute value by absorbing the sign into the constant. Many IB mark schemes expect you to express the final answer as y = f(x) where possible.
如果积分过程中出现对数，要记住 ln|y| = … 可化为 |y| = e^(…)，并通过将正负号吸收到常数中去掉绝对值。IB 评分标准通常期望你尽可能将最终答案写成 y = f(x) 的形式。
For example, consider dy/dx = 2xy. Separating gives ∫ (1/y) dy = ∫ 2x dx, so ln|y| = x² + C. Thus y = Ae^(x²), where A = ±e^C is the constant.
例如，考虑 dy/dx = 2xy。分离变量得 ∫ (1/y) dy = ∫ 2x dx，则 ln|y| = x² + C。于是 y = Ae^(x²)，其中 A = ±e^C 是常数。
4. First-Order Linear Equations and Integrating Factor | 一阶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). When it is not separable, the integrating factor method provides a systematic solution. The integrating factor is defined as μ(x) = e^(∫ P(x) dx).
一阶线性微分方程的标准形式为 dy/dx + P(x)y = Q(x)。当该方程不可分离变量时，积分因子法提供了一条系统性的求解路径。积分因子定义为 μ(x) = e^(∫ P(x) dx)。
Multiplying the entire equation by μ(x) transforms the left-hand side into the exact derivative of μ(x)y. The equation then becomes d/dx [μ(x)y] = μ(x)Q(x), which can be integrated directly.
用 μ(x) 乘以方程两边，便可将左侧转化为 μ(x)y 的全导数。方程变为 d/dx [μ(x)y] = μ(x)Q(x)，可直接积分求解。
Once integrated, solve for y and include the arbitrary constant. The method relies on the clever construction of μ(x), so always compute ∫ P(x) dx accurately, and you may omit the constant in this particular integral because it cancels out later.
积分后解出 y 并保留任意常数。该方法依靠 μ(x) 的巧妙构造，因此务必准确计算 ∫ P(x) dx，可在此处省略积分常数，因其最终会消去。
5. Steps for Integrating Factor Method | 积分因子法的步骤
Step 1: Write the DE in standard linear form, i.e., dy/dx + P(x)y = Q(x). If needed, divide through by the coefficient of dy/dx. Step 2: Compute μ(x) = e^(∫ P(x) dx).
第一步：将微分方程写成标准线性形式，即 dy/dx + P(x)y = Q(x)。如有必要，除以 dy/dx 的系数。第二步：计算 μ(x) = e^(∫ P(x) dx)。
Step 3: Multiply every term of the standard-form equation by μ(x). Step 4: Recognise that the left side is now (μ(x)y)′ and integrate both sides: μ(x)y = ∫ μ(x)Q(x) dx + C.
第三步：用 μ(x) 乘以标准形式方程中的每一项。第四步：识别出左侧即为 (μ(x)y)′，并两边积分：μ(x)y = ∫ μ(x)Q(x) dx + C。
Step 5: Isolate y by dividing by μ(x). Many candidates lose marks by incorrectly evaluating ∫ μ(x)Q(x) dx, so use integration by parts or substitution carefully and check your work.
第五步：除以 μ(x) 解出 y。许多考生因错误计算 ∫ μ(x)Q(x) dx 而失分，因此要谨慎使用分部积分或代换法，并检查计算过程。
Example: Solve dy/dx + (2/x)y = x². Here P(x) = 2/x, so ∫ P(x) dx = 2 ln|x|, μ(x) = x². Multiplying gives x² dy/dx + 2x y = x⁴. Thus (x² y)′ = x⁴, so x² y = (1/5)x⁵ + C, and y = (1/5)x³ + C/x².
示例：求解 dy/dx + (2/x)y = x²。这里 P(x) = 2/x，因此 ∫ P(x) dx = 2 ln|x|，μ(x) = x²。乘入得 x² dy/dx + 2x y = x⁴。于是 (x² y)′ = x⁴，故 x² y = (1/5)x⁵ + C，最终 y = (1/5)x³ + C/x²。
6. Homogeneous Equations of the Form dy/dx = f(y/x) | 齐次方程 dy/dx = f(y/x)
A first-order DE is called homogeneous if it can be expressed as dy/dx = F(y/x). Such equations are solved by the substitution y = vx, where v is a function of x. Then dy/dx = v + x dv/dx, and the original equation becomes separable in v and x.
若一阶微分方程可表示为 dy/dx = F(y/x) 的形式，则称其为齐次方程。这类方程可通过代换 y = vx 求解，其中 v 是 x 的函数。此时 dy/dx = v + x dv/dx，原方程化为关于 v 和 x 的可分离变量方程。
After substituting, you obtain v + x dv/dx = F(v). This simplifies to x dv/dx = F(v) – v, which can be separated as dv/(F(v)-v) = dx/x. Integrate both sides, then substitute back v = y/x to obtain the solution in terms of x and y.
代换后得到 v + x dv/dx = F(v)。化简为 x dv/dx = F(v) – v，可分离为 dv/(F(v)-v) = dx/x。两边积分后，再用 v = y/x 回代，得到用 x 和 y 表示的解。
Be careful with algebraic manipulation when separating. It is easy to misplace a minus sign or mishandle the integral of 1/x. Always check your final expression by differentiating.
分离变量时要注意代数处理，很容易搞错符号或处理 1/x 的积分时出错。最终得到的表达式一定要求导验证。
7. Second-Order Homogeneous Linear Equations with Constant Coefficients | 二阶常系数齐次线性方程
A second-order linear homogeneous DE with constant coefficients takes the form a y″ + b y′ + c y = 0, where a, b, c are constants. Its behaviour is entirely determined by the characteristic (auxiliary) equation a r² + b r + c = 0.
二阶常系数齐次线性微分方程的形式为 a y″ + b y′ + c y = 0，其中 a, b, c 为常数。其性态完全由特征（辅助）方程 a r² + b r + c = 0 决定。
To derive this, we assume a trial solution of the form y = e^(rx). Substituting gives e^(rx)(a r² + b r + c) = 0, so the exponential factor forces the quadratic to be zero.
为了导出它，我们假设试解 y = e^(rx)。代入得到 e^(rx)(a r² + b r + c) = 0，因为指数函数非零，故二次式必为零。
The solutions r₁ and r₂ dictate the form of the general solution. For IB and AQA, you must be able to handle three distinct cases: two distinct real roots, a repeated real root, and complex conjugate roots.
解 r₁ 和 r₂ 决定了通解的形式。在 IB 与 AQA 考试中，你必须能处理三种不同情形：两个不等实根、一个重实根以及共轭复根。
8. The Characteristic Equation and Three Cases | 特征方程与三种情况
Case 1: Two distinct real roots r₁ ≠ r₂. The general solution is y = Ae^(r₁x) + Be^(r₂x), where A and B are arbitrary constants. This occurs when the discriminant b² – 4ac > 0.
情形一：两个不等实根 r₁ ≠ r₂。 通解为 y = Ae^(r₁x) + Be^(r₂x)，其中 A 和 B 为任意常数。这发生在判别式 b² – 4ac > 0 时。
Case 2: Repeated real root r = r₁ = r₂. The solution is y = (A + Bx)e^(rx). The extra x factor arises because one solution alone cannot span the two-dimensional solution space.
情形二：重实根 r = r₁ = r₂。 解为 y = (A + Bx)e^(rx)。多出的 x 因子是因为仅有一个解无法生成二维解空间。
Case 3: Complex conjugate roots r = α ± iβ. The general solution is y = e^(αx)(A cos βx + B sin βx). Use Euler’s formula and the fact that the real and imaginary parts are independent solutions.
情形三：共轭复根 r = α ± iβ。 通解为 y = e^(αx)(A cos βx + B sin βx)。利用欧拉公式以及实部和虚部都是独立的解这一事实。
Always write the auxiliary equation clearly and compute its roots exactly or to the required precision. In exam settings, surd forms are often required.
务必清晰地写出辅助方程，并精确计算根或按题目要求保留根号形式。考试中常要求保留根式。
9. Initial and Boundary Conditions | 初始条件与边界条件
General solutions contain arbitrary constants. To find a particular solution, you need extra information, such as initial values (e.g., y(0) = 1, y′(0) = 2) or boundary values. These conditions are substituted into the general solution and its derivatives to form simultaneous equations for the constants.
通解包含任意常数。要得到特解，需要额外信息，如初始值（例如 y(0) = 1, y′(0) = 2）或边值条件。将这些条件代入通解及其导数，可以建立关于常数的联立方程。
For a first-order DE, one condition is sufficient. For a second-order DE, you need two conditions to determine both constants uniquely. Be systematic: first write the general solution, then differentiate if necessary, and substitute the given x and y values.
对于一阶微分方程，一个条件就足够了。对于二阶方程，需要两个条件才能唯一确定两个常数。要有条理：先写出通解，必要时求导，然后代入给定的 x 和 y 值。
Check that your particular solution satisfies the original DE and the conditions. A quick differentiation and substitution can prevent algebraic errors that would otherwise cost valuable marks.
检查所得特解是否满足原方程和给定条件。快速求导并代入可避免代数错误，免于失分。
10. Applications: Growth, Decay, and Cooling | 应用：增长、衰减与冷却定律
Many exam questions model real-world behaviour. Exponential growth and decay are governed by dy/dt = ky, whose general solution is y = Ce^(kt). If k > 0 it is growth; if k < 0 it is decay. The half-life or doubling time can be found by setting y = ½ y₀ or 2 y₀.
许多试题都会模拟现实世界的行为。指数增长与衰减由 dy/dt = ky 控制，其通解为 y = Ce^(kt)。若 k > 0 则为增长，若 k < 0 则为衰减。通过设 y = ½ y₀ 或 2 y₀ 可求出半衰期或倍增时间。
Newton’s law of cooling states that the rate of change of temperature is proportional to the difference between the object’s temperature T and the ambient temperature Tₐ: dT/dt = -k(T – Tₐ). The solution is T = Tₐ + (T₀ – Tₐ)e^(-kt).
牛顿冷却定律指出，温度变化率与物体温度 T 和环境温度 Tₐ 之差成正比：dT/dt = -k(T – Tₐ)。其解为 T = Tₐ + (T₀ – Tₐ)e^(-kt)。
Always define your variables and constants clearly in contextual questions. Translate ‘initially, the temperature is 80°C’ into T(0) = 80, and use this to find the integration constant. Units and interpretation are part of the final answer.
在情景题中务必清晰定义变量和常数。将“初始温度为 80°C”转化为 T(0) = 80，并利用它求出积分常数。单位和解释也是最终答案的一部分。
11. Exam Tips and Common Mistakes | 考试技巧与常见错误
Mistake 1: Forgetting the constant of integration. Every indefinite integration in the solution process must produce a ‘+ C’. Losing this constant changes the general solution into a particular one and will cause you to lose marks.
错误一：忘记积分常数。求解过程中每一步不定积分都必须产生一个“+ C”。漏掉这个常数会将通解变成特解，从而失分。
Mistake 2: Misidentifying the form of a DE. Before applying a method, always check if the equation is separable, linear, or homogeneous. Writing down the standard form helps avoid choosing the wrong technique.
错误二：误判微分方程的类型。在运用方法前，始终要检查方程是可分离、线性还是齐次的。写出标准形式有助于避免选错方法。
Mistake 3: Errors when integrating the integrating factor or when solving the auxiliary equation. Double-check algebraic expansions, factorisations, and sign conventions. In second-order DEs, mis-copying the sign of b in b² – 4ac is extremely common.
错误三：积分因子积分或求解辅助方程时出错。反复检查代数展开、因式分解和符号规则。在二阶微分方程中，将判别式 b² – 4ac 中 b 的符号写错极为常见。
Tip: Always verify your final solution by substituting back into the original equation. Even a quick mental check can catch inconsistent terms. Also, draw on the context for application problems to confirm that the solution makes physical sense.
建议：始终将最终解回代到原方程中进行验证。即使是快速心算也能发现不一致的项。此外，在应用题中要根据上下文判断解是否具有物理意义。
12. Summary of Solution Methods | 解法总结
A concise overview of the methods covered: separable equations use direct integration after separating variables; first-order linear equations use the integrating factor μ(x) = e^(∫ P(x) dx); homogeneous equations use the substitution y = vx; second-order constant-coefficient homogeneous equations use the characteristic equation and three case-based general solutions.
所讲方法的简明概览：可分离变量方程在分离变量后直接积分；一阶线性方程使用积分因子 μ(x) = e^(∫ P(x) dx)；齐次方程使用代换 y = vx；二阶常系数齐次方程使用特征方程以及基于三种情况的通解。
For applications, translate the verbal description into a DE with an initial condition, solve, and then interpret the constants. Practice recognising the structure quickly so that you can select the efficient method under timed conditions.
对于应用题，将文字描述转化为带有初始条件的微分方程，求解后再对常数进行解释。要通过练习快速识别结构，以便在限时考试中选出最高效的方法。
Keep a formula sheet handy while revising, but work towards memorising the standard forms and solution templates. Mastery of differential equations not only secures a significant portion of the calculus marks but also strengthens your overall problem-solving skills.
复习时手边备一份公式表，但要努力熟记标准形式和求解模板。掌握好微分方程不仅能帮你稳稳拿下微积分中的相当部分分数，还能提升整体解题能力。
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