Differential Equations Exam Essentials | 微分方程考点精讲

📚 Differential Equations Exam Essentials | 微分方程考点精讲

Differential equations are a core topic in both IB Higher Level Mathematics: Analysis and Approaches and CCEA A-Level Mathematics. They allow us to model dynamic systems and solve a wide range of problems, from population growth to mechanical vibrations. In this exam-focused guide, we will systematically review the essential techniques, common pitfalls, and application strategies required for success.

微分方程是 IB 高水平数学：分析与方法以及 CCEA A-Level 数学的核心内容。通过微分方程，我们可以对动态系统建模，解决从人口增长到机械振动等诸多实际问题。本文将从考点出发，系统梳理求解技巧、常见错误和应用策略，帮助考生高效备考。

1. Introduction to Differential Equations | 微分方程简介

A differential equation (DE) is an equation that contains derivatives of an unknown function. For example, dy/dx = 3x² is a simple first-order DE. In both IB and CCEA syllabi, you need to be able to classify, solve, and interpret differential equations.

微分方程是包含未知函数导数的方程。例如 dy/dx = 3x² 就是一个简单的一阶微分方程。在 IB 和 CCEA 考纲中，学生需要掌握分类、求解和解释微分方程的能力。

Differential equations arise whenever a rate of change is related to the quantity itself. Recognising the type of DE is the first step towards choosing the appropriate solution method.

只要变化率与量本身有关，就会产生微分方程。识别方程的类型是选择适切求解方法的第一步。

2. Classification and Order | 分类与阶数

The order of a differential equation is the highest derivative that appears. For instance, d²y/dx² + 2 dy/dx + y = 0 is a second-order DE. A differential equation is linear if the dependent variable and its derivatives appear only to the first power and are not multiplied together.

微分方程的阶数是方程中出现的最高阶导数。例如 d²y/dx² + 2 dy/dx + y = 0 是一个二阶微分方程。如果因变量及其导数只以一次幂出现且互不乘积，则称该方程为线性微分方程。

In the IB and CCEA syllabi, you are primarily concerned with first-order equations (separable and linear) and second-order linear equations with constant coefficients. Recognising the order and linearity immediately narrows down the solving strategies.

在 IB 和 CCEA 课程中，主要关注一阶方程（可分离和线性）以及常系数二阶线性方程。识别阶数和线性性质能立即缩小求解策略的范围。

3. First Order Separable Equations | 一阶可分离变量方程

An equation of the form dy/dx = f(x)g(y) can be solved by separation. Rearrange so that all y terms are on one side with dy and all x terms on the other with dx, then integrate both sides.

形如 dy/dx = f(x)g(y) 的方程可通过分离变量法求解。将所有含 y 的项及 dy 移到一边，所有含 x 的项及 dx 移到另一边，然后两边积分。

Steps: 1. Write dy/dx = f(x)g(y). 2. Separate: (1/g(y)) dy = f(x) dx. 3. Integrate: ∫(1/g(y)) dy = ∫f(x) dx + C. 4. Solve for y if possible, or leave the solution in implicit form.

步骤：1. 写出 dy/dx = f(x)g(y)。2. 分离：(1/g(y)) dy = f(x) dx。3. 积分：∫(1/g(y)) dy = ∫f(x) dx + C。4. 如可能，解出显式 y，或者保留隐式解。

Example: Solve dy/dx = x/y. Separation gives y dy = x dx. Integrating yields ½ y² = ½ x² + C, so y² = x² + 2C, often written as y² = x² + k. This describes a family of hyperbolas.

例子：解 dy/dx = x/y。分离得 y dy = x dx。积分得 ½ y² = ½ x² + C，即 y² = x² + k。这表示一族双曲线。

4. First Order Linear Equations and Integrating Factors | 一阶线性方程与积分因子

A first-order linear differential equation can be written in standard form: dy/dx + P(x)y = Q(x). The integrating factor (IF) is μ(x) = e^(∫P(x)dx). Multiplying both sides by μ transforms the left-hand side into the derivative of μ y.

一阶线性微分方程可写成标准形式：dy/dx + P(x)y = Q(x)。积分因子 μ(x) = e^(∫P(x)dx)。将方程两边乘以 μ，左边便化为 μ y 的导数。

The general solution is y = (1/μ(x)) [ ∫ μ(x)Q(x) dx + C ]. This technique is required in IB HL and CCEA A-Level, so memorising the formula is crucial.

通解为 y = (1/μ(x)) [ ∫ μ(x)Q(x) dx + C ]。IB HL 和 CCEA A-Level 都要求掌握这一方法，因此牢记公式至关重要。

Example: Solve dy/dx + (2/x)y = x. Here P(x)=2/x, μ = e^(∫2/x dx) = e^(2 ln x) = x². Multiply: d/dx (x² y) = x³. Integrate: x² y = ¼ x⁴ + C, so y = ¼ x² + C/x².

例子：解 dy/dx + (2/x)y = x。此处 P(x)=2/x, μ = e^(∫2/x dx) = x²。相乘后：d/dx (x² y) = x³。积分：x² y = ¼ x⁴ + C，所以 y = ¼ x² + C/x²。

5. Homogeneous First Order Equations | 一阶齐次方程（选讲）

A first-order differential equation of the form dy/dx = F(y/x) is called homogeneous. The substitution v = y/x, so y = vx and dy/dx = v + x dv

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