A-Level CIE Mathematics: Mastering Differential Equations | A-Level CIE 数学：微分方程考点精讲

📚 A-Level CIE Mathematics: Mastering Differential Equations | A-Level CIE 数学：微分方程考点精讲

In A-Level CIE Mathematics (9709), differential equations form a vital part of the Pure Mathematics 3 syllabus. They allow us to model dynamic systems involving rates of change, from population growth to mechanical oscillations. This revision guide covers all key concepts, solution techniques, and common exam applications, ensuring you are fully prepared for Paper 3 questions.

在 A-Level CIE 数学 (9709) 中，微分方程是纯数学 3 模块的重要组成部分。它们使我们能够对涉及变化率的动态系统进行建模，从人口增长到机械振动。本复习指南涵盖所有核心概念、解题技巧和常见考试应用，确保你为 P3 试题做好充分准备。

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

A differential equation is an equation that involves an unknown function and its derivatives. For example, dy/dx = 2xy is a first-order differential equation because it contains the first derivative dy/dx.

微分方程是包含未知函数及其导数的方程。例如，dy/dx = 2xy 是一个一阶微分方程，因为它包含一阶导数 dy/dx。

In the CIE syllabus, you will mainly encounter first-order ordinary differential equations where the derivative is expressed in terms of x and y, and you need to find the original relationship between these variables.

在 CIE 大纲中，你主要会遇到一阶常微分方程，其中导数用 x 和 y 表示，你需要找出这些变量之间的原始关系。

The notation y’ or f'(x) is sometimes used, but CIE questions usually write dy/dx explicitly to avoid ambiguity.

有时会使用记号 y’ 或 f'(x)，但 CIE 的题目通常明确写出 dy/dx 以避免歧义。

2. Order and Degree of a Differential Equation | 微分方程的阶与次

The order of a differential equation is the highest derivative present. For instance, d²y/dx² + 3 dy/dx = 0 is of order 2. The degree is the power of the highest derivative after the equation has been made free from radicals and fractions in derivatives.

微分方程的阶是出现的最高阶导数。例如，d²y/dx² + 3 dy/dx = 0 是二阶的。次是指方程中最高阶导数的幂次，前提是方程已消除了根号和分式形式的导数。

In CIE exams, almost all questions focus on first-order, first-degree equations, where the highest derivative appears to the power 1, such as dy/dx = f(x, y).

在 CIE 考试中，几乎所有的题目都集中在一阶一次方程，即最高阶导数的次数为 1，例如 dy/dx = f(x, y)。

It is essential to recognise the type of equation before attempting a solution method because higher-order equations require different techniques not covered in P3.

在尝试求解方法之前，识别方程的类型至关重要，因为高阶方程需要不同的技巧，而 P3 模块不涉及这些。

3. General Solution vs Particular Solution | 通解与特解

The general solution of a differential equation contains an arbitrary constant (e.g., C). It represents a family of curves that all satisfy the differential equation.

微分方程的通解包含一个任意常数（如 C）。它代表一族曲线，所有曲线都满足该微分方程。

A particular solution is obtained when an initial condition (e.g., y = 1 when x = 0) is used to determine the constant. This singles out one specific curve from the family.

当使用初始条件（例如，当 x = 0 时 y = 1）确定常数后，就得到特解。这从曲线族中选出一条特定的曲线。

You must always include the constant of integration when integrating and then substitute the given condition to find its value. Never forget +C!

积分时必须始终包含积分常数，然后代入给定的条件求其数值。千万不要忘记 +C！

4. Separable Differential Equations | 可分离变量微分方程

A first-order differential equation is separable if it can be written in the form dy/dx = f(x) g(y). That is, the right-hand side is a product of a function of x and a function of y.

如果一个一阶微分方程可以写成 dy/dx = f(x) g(y) 的形式，那么它就是可分离的。即右边是 x 的函数与 y 的函数的乘积。

For example, dy/dx = 3x² / y is separable: f(x) = 3x², g(y) = 1/y. Another example: dy/dx = xy + x is also separable because it can be factorised as x(y+1), giving f(x)=x, g(y)=y+1.

例如，dy/dx = 3x² / y 是可分离的：f(x) = 3x², g(y) = 1/y。另一个例子：dy/dx = xy + x 也是可分离的，因为它可以因式分解为 x(y+1)，即 f(x)=x, g(y)=y+1。

Recognising separable equations is the key first step in CIE P3 differential equations; always check if the expression can be factorised into a product of a function of x and a function of y.

识别可分离方程是 CIE P3 微分方程的关键第一步；始终检查表达式是否可以因式分解为一个仅含 x 的函数与一个仅含 y 的函数的乘积。

5. Solving First-Order Separable Equations: Step-by-Step | 一阶可分离方程的求解步骤

Step 1: Rewrite the equation to isolate the derivative: dy/dx = f(x) g(y). If necessary, factorise to achieve this form.

步骤 1：改写方程，分离导数：dy/dx = f(x) g(y)。必要时进行因式分解以达到这种形式。

Step 2: Separate variables by dividing both sides by g(y) and multiplying by dx: (1/g(y)) dy = f(x) dx. Ensure g(y) ≠ 0.

步骤 2：通过两边同除以 g(y) 并乘以 dx 来分离变量：(1/g(y)) dy = f(x) dx。确保 g(y) ≠ 0。

Step 3: Integrate both sides: ∫ (1/g(y)) dy = ∫ f(x) dx. Use standard integration techniques and add a single constant of integration C on one side (usually the x-side).

步骤 3：两边积分：∫ (1/g(y)) dy = ∫ f(x) dx。使用标准积分方法，在一边（通常是 x 的一边）加上一个积分常数 C。

Step 4: Perform the integration carefully, including any necessary substitutions, and then simplify.

步骤 4：仔细进行积分，包括任何必要的换元，然后化简。

Step 5: If possible, solve for y explicitly to give the solution in the form y = … otherwise leave it in implicit form unless the question requests an explicit expression.

步骤 5：如果可能，将 y 表示为显函数，即 y = … 的形式；否则，除非题目要求给出显式表达式，可以保留隐式形式。

Example: Solve dy/dx = 2xy, given y(0) = 3.

例题：解 dy/dx = 2xy，已知 y(0) = 3。

Separate: (1/y) dy = 2x dx, then integrate: ln|y| = x² + C. Substitute y=3 at x=0: ln3 = C. Hence, ln|y| = x² + ln3 → y = 3 eˣ².

分离变量：(1/y) dy = 2x dx，然后积分：ln|y| = x² + C。代入 x=0 时 y=3：ln3 = C。因此，ln|y| = x² + ln3 → y = 3 eˣ²。

6. Initial Conditions and Finding the Constant | 初始条件与常数求解

After obtaining the general solution, use the given initial condition, e.g., y(x₀) = y₀, to substitute into the solution. Solve the resulting equation for C.

得到通解后，使用给定的初始条件，例如 y(x₀) = y₀，代入解中。求解所得关于 C 的方程。

If the general solution is implicit, substitute x₀ and y₀ directly and rearrange to find C. If explicit, plug in the numbers.

如果通解是隐式的，直接代入 x₀ 和 y₀ 并整理求出 C。如果是显式的，代入数值即可。

Common initial conditions include boundary values, like y = 2 when x = 0, or a known point through which the curve passes. Always present the final particular solution clearly.

常见的初始条件包括边界值，比如 x = 0 时 y = 2，或者已知曲线经过的某个点。始终清晰地给出最终的特解。

For instance, if you find ln|y| = ½x² + C and know (0, 4) lies on the curve, then ln4 = 0 + C → C = ln4, so y = 4 e^(½x²).

例如，如果你得到 ln|y| = ½x² + C 且知道点 (0, 4) 在曲线上，那么 ln4 = 0 + C → C = ln4，因此 y = 4 e^(½x²)。

7. Modelling with Differential Equations: Growth and Decay | 微分方程建模：增长与衰减

The differential equation dP/dt = kP models exponential growth (k > 0) or decay (k < 0), where P is the population or quantity at time t, and k is the relative growth rate.

微分方程 dP/dt = kP 模拟指数增长 (k > 0)

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A-Level CIE Mathematics: Mastering Differential Equations | A-Level CIE 数学：微分方程 考点精讲