The Concept of a Differential Equation: Order and Linearity | 微分方程的概念：阶与线性

📚 The Concept of a Differential Equation: Order and Linearity | 微分方程的概念：阶与线性

A differential equation links an unknown function to its derivatives and is central to modelling rates of change in AQA A-Level Mathematics. This article explains what a differential equation is, how to classify it by order and linearity, and why these classifications guide the choice of solution method.

微分方程将未知函数与其导数联系起来，是 AQA A-Level 数学中建立变化率模型的核心工具。本文解释什么是微分方程、如何按阶和线性进行分类，以及这些分类为何能指导求解方法的选择。

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

A differential equation is an equation involving an unknown function and one or more of its derivatives. For example, dy/dx = 3x² is a differential equation because it contains dy/dx. The aim is to recover the original function y = f(x) from information about its rate of change.

微分方程是包含未知函数及其一个或多个导数的方程。例如，dy/dx = 3x² 就是一个微分方程，因为它含有 dy/dx。目标是根据变化率的信息求出原函数 y = f(x)。

In A-Level problems, the unknown function is usually y = f(x) and the independent variable is x. Differential equations can be written using dy/dx, d²y/dx², or prime notation f'(x), f”(x).

在 A-Level 问题中，未知函数通常是 y = f(x)，自变量是 x。微分方程可以使用 dy/dx、d²y/dx² 或撇号记法 f'(x)、f”(x) 表示。

2. Notation and Basic Terminology | 记号与基本术语

The derivative dy/dx represents the first derivative of y with respect to x. The second derivative d²y/dx² represents the rate of change of dy/dx. In an equation, the order is determined by the highest derivative present, and linearity depends on how y and its derivatives appear.

导数 dy/dx 表示 y 对 x 的一阶导数。二阶导数 d²y/dx² 表示 dy/dx 的变化率。在方程中，阶由出现的最高阶导数决定，而线性取决于 y 及其导数出现的方式。

Solutions to a differential equation are functions, not single numbers. A general solution contains an arbitrary constant, while a particular solution satisfies an extra condition such as y(1) = 4.

微分方程的解是函数，而不是单个数值。通解含有一个任意常数，而特解则满足额外条件，例如 y(1) = 4。

3. Order of a Differential Equation | 微分方程的阶

The order of a differential equation is the order of the highest derivative that appears in it. First-order equations contain dy/dx but no higher derivatives; second-order equations contain d²y/dx² but no third or higher derivatives.

微分方程的阶是方程中出现的最高阶导数的阶数。一阶方程含有 dy/dx，但不含更高阶导数；二阶方程含有 d²y/dx²，但不含三阶或更高阶导数。

dy/dx + 2y = eˣ (first order)

d²y/dx² + 5 dy/dx + 6y = 0 (second order)

Identifying the order is usually the first step because different orders require different solution strategies. In AQA papers, most differential equations you solve are first order, but you must still recognise second-order equations.

识别阶通常是第一步，因为不同阶需要不同的求解策略。在 AQA 试卷中，你要求解的大多数微分方程是一阶的，但你仍必须识别二阶方程。

4. Linearity: The Core Idea | 线性：核心思想

A differential equation is linear if the unknown function y and all its derivatives appear only to the first power, are not multiplied together, and are not inside nonlinear functions such as sin, cos, e^y, or y². For example, dy/dx + 3y = x is linear because y and dy/dx appear separately and only to power 1.

如果未知函数 y 及其所有导数只以一次幂出现、彼此不相乘，并且不在 sin、cos、e^y 或 y² 等非线性函数内部，则微分方程是线性的。例如，dy/dx + 3y = x 是线性的，因为 y 和 dy/dx 分别出现且只是一次幂。

Linearity is about the dependent variable y, not the independent variable x. Terms such as x², sin x, or eˣ on the right-hand side do not make the equation nonlinear as long as y and its derivatives are linear.

线性是针对因变量 y，而不是自变量 x。只要 y 及其导数是线性的，右边的 x²、sin x 或 eˣ 等项并不会使方程变为非线性。

5. Linear Differential Equations | 线性微分方程

A first-order linear differential equation can be written in the standard form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x only. This form is important because AQA methods such as integrating factors rely on it.

一阶线性微分方程可以写成标准形式 dy/dx + P(x)y = Q(x)，其中 P(x) 和 Q(x) 只是 x 的函数。这个形式很重要，因为 AQA 的积分因子等方法依赖于它。

dy/dx + P(x)y = Q(x)

For example, dy/dx + 2xy = x³ is linear with P(x) = 2x and Q(x) = x³. Linear equations have the useful property that solutions can be combined in certain ways, though at A-Level you will focus on finding one solution.

例如，dy/dx + 2xy = x³ 是线性的，其中 P(x) = 2x，Q(x) = x³。线性方程有一个有用性质：解可以按特定方式组合，不过在 A-Level 中你主要关注求出一个解。

6. Nonlinear Differential Equations |

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