Differential Equations Revision Guide | 微分方程考点精讲

📚 Differential Equations Revision Guide | 微分方程考点精讲

In IGCSE Additional Mathematics, differential equations let us model situations where the rate of change of a quantity relates to the quantity itself. They appear in population growth, radioactive decay, cooling processes and kinematics. The exam expects you to solve first‑order separable equations, interpret boundary conditions and translate real‑world problems into differential equations.

在 IGCSE 附加数学中，微分方程帮助我们建立「量的变化率与量本身有关」的模型。这类问题常出现在人口增长、放射性衰变、冷却过程和运动学中。考试要求你掌握一阶可分离变量方程的求解、解读边界条件，并能把实际问题转化为微分方程。

1. What is a Differential Equation? | 什么是微分方程？

A differential equation involves an unknown function and its derivatives. For example, dy/dx = 3x² is a simple differential equation whose solution is y = x³ + C.

微分方程包含未知函数及其导数。例如 dy/dx = 3x² 是一个简单的微分方程，其解为 y = x³ + C。

The order of a differential equation is determined by the highest derivative that appears. At IGCSE you only meet first‑order equations, where the highest derivative is dy/dx.

微分方程的阶数由出现的最高阶导数决定。IGCSE 阶段只涉及一阶方程，即最高导数为 dy/dx。

2. Notation and Terminology | 符号与术语

We usually write the derivative as dy/dx or y’. For a rate of change with respect to time t we use dx/dt, dv/dt, etc. The general solution contains one arbitrary constant C.

我们通常把导数记作 dy/dx 或 y’。对于随时间 t 变化的变化率，常用 dx/dt、dv/dt 等。通解包含一个任意常数 C。

A particular solution is obtained when we use an extra condition – such as the value of y when x = 0 – to find C.

特解则需要利用额外条件（例如当 x = 0 时 y 的值）求出常数 C。

3. Separable Differential Equations | 可分离变量的微分方程

A first‑order equation is called separable if it can be written in the form dy/dx = f(x)·g(y). The variables x and y can be separated onto different sides of the equation.

若一阶方程可以写成 dy/dx = f(x)·g(y) 的形式，就称为可分离变量方程。此时变量 x 和 y 可以分别移到等式的两边。

For instance, dy/dx = 2xy is separable because we can rewrite it as (1/y)dy = 2x dx. This rearrangement is the key step to solving the equation.

例如 dy/dx = 2xy 就是可分离变量的，因为我们可以将其改写为 (1/y)dy = 2x dx。这种重排是求解方程的关键步骤。

4. Step‑by‑Step Separation of Variables | 分离变量法步骤

To solve dy/dx = f(x)g(y), first rewrite it as 1/g(y) dy = f(x) dx. Then integrate both sides: ∫ 1/g(y) dy = ∫ f(x) dx. After integration, add the constant of integration on one side.

求解 dy/dx = f(x)g(y) 时，先改写成 1/g(y) dy = f(x) dx。然后两边积分：∫ 1/g(y) dy = ∫ f(x) dx。积分后，在等式一侧加上积分常数。

Finally, rearrange the result to express y explicitly in terms of x, unless a question allows an implicit form.

最后将结果整理成 y 关于 x 的显函数的表达式，除非题目允许保留隐函数形式。

5. General Solution with the Constant of Integration | 带积分常数的通解

After integration you obtain an equation involving x, y and a constant C. This is the general solution – it represents a family of curves, each corresponding to a different value of C.

积分之后，你会得到一个含有 x、y 和常数 C 的方程。这就是通解，它表示一族曲线，每条曲线对应一个不同的 C 值。

Example: dy/dx = 3x² → y = x³ + C

There are infinitely many curves, all with the same shape, shifted vertically.

这族曲线形状相同，只在竖直方向上平移。

6. Using Initial Conditions to Find Particular Solution | 用初始条件求特解

A particular solution is found by substituting a known pair (x₀, y₀) into the general solution. This gives a numerical value for C.

将已知的一组值 (x₀, y₀) 代入通解，就可以求出 C 的数值，从而得到特解。

In IGCSE questions the condition is often given as “when x = …, y = …” or “at time t = 0, …”. Always use these to determine C before writing the final answer.

IGCSE 题目中常会给出条件“当 x = … 时 y = …”或“在 t = 0 时 ……”。一定要先用这些条件确定常数 C，再写出最终答案。

7. Exponential Growth Model | 指数增长模型

A common differential equation for growth is dx/dt = kx, where k > 0. The rate of change is directly proportional to the current amount.

常见的增长微分方程是 dx/dt = kx，其中 k > 0。变化率与当前数量成正比。

Separating variables gives ∫ (1/x)dx = ∫ k dt, leading to ln|x| = kt + C. Writing in exponential form yields x = A eᵏᵗ, where A = eᶜ.

分离变量得到 ∫ (1/x)dx = ∫ k dt，即 ln|x| = kt + C。写成指数形式为 x = A eᵏᵗ，其中 A = eᶜ。

This model is used for unlimited population growth, compound interest (with continuous compounding) and similar situations.

该模型可用于无限制的人口增长、连续复利等情境。

8. Exponential Decay Model | 指数衰减模型

When k is negative, the equation dx/dt = –kx (with k > 0) models decay. The solution is x = A e⁻ᵏᵗ, where A is the initial amount at t = 0.

当 k 为负值时，方程 dx/dt = –kx（k > 0）描述衰减过程。其解为 x = A e⁻ᵏᵗ，其中 A 是 t = 0 时的初始量。

Radioactive decay, falling temperature difference and drug concentration in the bloodstream often follow this law. The half‑life is the time taken for the quantity to halve.

放射性衰变、温度差的下降以及血液中药物浓度通常遵循这一定律。半衰期是指数量减半所需的时间。

9. Differential Equations in Kinematics | 运动学中的微分方程

In kinematics, velocity v is the rate of change of displacement s with time: v = ds/dt. Acceleration a is the rate of change of velocity: a = dv/dt, which is also d²s/dt².

在运动学中，速度 v 是位移 s 对时间 t 的变化率：v = ds/dt。加速度 a 是速度的变化率：a = dv/dt，也是 d²s/dt²。

If a is given as a function of t, you integrate once to find v, and again to find s. If a is expressed as a function of v or s, you often set up a separable differential equation, such as dv/dt = f(v).

如果加速度 a 表示为 t 的函数，积分一次可得 v，再积分一次得 s。如果 a 表示为 v 或 s 的函数，通常需要建立可分离变量的微分方程，例如 dv/dt = f(v)。

Example: dv/dt = –kv → v = v₀ e⁻ᵏᵗ

10. Forming Differential Equations from a Description | 从文字描述建立微分方程

Read the wording carefully: “the rate of increase of P is proportional to P” translates to dP/dt = kP. “The rate of decrease is proportional to the square root of Q” gives dQ/dt = –k√Q.

仔细阅读题干：“P 的增加速率与 P 成正比”可以翻译为 dP/dt = kP。“减少速率与 Q 的平方根成正比”则对应 dQ/dt = –k√Q。

Always state the proportional constant k with the correct sign based on increase or decrease. Use the given data to find k after solving the differential equation.

一定要根据增加或减少的情境为比例常数 k 配上正确的符号。求解微分方程后，再使用给定数据求出 k 的具体值。

11. Common Mistakes to Avoid | 常见错误避免

Forgetting the constant of integration: Always add +C immediately after integration; losing it will cost marks.
忘记加积分常数：积分后必须立即加上 +C，丢掉常数会被扣分。
Incorrect separation: Check that the RHS is truly a product of a function of x and a function of y. dy/dx = x+y is not separable.
分离不全：确认等式右边确实是关于 x 的函数和关于 y 的函数的乘积。dy/dx = x+y 不是可分离变量的。
Misuse of modulus: When integrating 1/y you obtain ln|y|. You can drop the absolute value only if you know y is always positive.
绝对值误用：对 1/y 积分得到的是 ln|y|。只有在确知 y 恒为正时才可以省去绝对值符号。
Arithmetic errors in exponentiation: e^(ln|y|+C) simplifies to eᶜ·|y|, not eˡⁿ|ʸ| + eᶜ.
指数运算错误：e^(ln|y|+C) 化简为 eᶜ·|y|，而不是 eˡⁿ|ʸ| + eᶜ。

12. Exam Tips for Top Marks | 高分应试技巧

Show clear steps: Write the separation, the two integrals with correct notation, the constant of integration, and the simplification.
写出清晰步骤：写出分离过程、两边积分（带上正确的积分记号）、积分常数以及化简过程。
Use initial conditions early: As soon as you have the general solution, substitute the condition to find C or k. This avoids messy algebra later.
尽早使用初始条件：得到通解后立即代入条件求 C 或 k，可以避免后续繁琐的代数演算。
Re‑read the question: Make sure your final answer is in the form requested – explicit y = f(x), or simplified as far as possible, or with k calculated to the appropriate precision.
再次审题：确保最终答案符合题目要求的格式——显函数形式 y = f(x)、尽可能简化、或按适当精度给出 k 值。
Know the standard integrals: ∫ (1/x) dx = ln|x|, ∫ xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ –1), ∫ eᵃˣ dx = (1/a)eᵃˣ.
熟记标准积分：∫ (1/x) dx = ln|x|，∫ xⁿ dx = xⁿ⁺¹/(n+1)（n ≠ –1），∫ eᵃˣ dx = (1/a)eᵃˣ。

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Differential Equations Revision Guide | 微分方程考点精讲