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 is a mathematical equation that relates a function to its derivatives. It describes how a quantity changes over time or space, and it lies at the heart of modelling real-world phenomena, from population growth to radioactive decay. For A-Level Mathematics, the essential first step is to understand what a differential equation is, and how to classify it by its order and by whether it is linear or non-linear.

微分方程是将一个函数与其导数联系起来的数学方程。它描述了一个量如何随时间或空间变化，是模拟现实世界现象的核心工具，从人口增长到放射性衰变皆是如此。在 A-Level 数学中，关键的起步步骤是理解什么是微分方程，以及如何通过其阶数、线性或非线性对其进行分类。

1. Defining a Differential Equation | 微分方程的定义

A differential equation is an equation involving an unknown function and one or more of its derivatives. For example, if y = f(x), then an equation such as dy/dx = 3x² + 2 is a differential equation because it contains the derivative dy/dx. The goal is usually to find the function y that satisfies the equation for all relevant values of x.

微分方程是包含一个未知函数及其一个或多个导数的方程。例如，若 y = f(x)，则像 dy/dx = 3x² + 2 这样的方程就是微分方程，因为它含有导数 dy/dx。我们的目标通常是找到在所有相关 x 值下都满足该方程的函数 y。

Differential equations arise naturally whenever we know the rate of change of a quantity. For instance, Newton’s law of cooling states that the rate at which an object cools is proportional to the temperature difference between the object and its surroundings. This statement translates directly into a differential equation.

当我们知道某个量的变化率时，微分方程便会自然出现。例如，牛顿冷却定律指出，物体冷却的速率与物体和周围环境之间的温度差成正比。这一表述可直接转化为一个微分方程。

2. The 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. If the highest derivative is the first derivative, the equation is called a first-order differential equation. If the highest derivative is the second derivative, it is a second-order differential equation, and so on.

微分方程的阶是方程中出现的最高阶导数的阶数。如果最高导数为一级导数，则称为一阶微分方程；如果最高导数为二级导数，则为二阶微分方程，依此类推。

dy/dx + 2y = eˣ → First-order (highest derivative is dy/dx)

dy/dx + 2y = eˣ → 一阶（最高导数为 dy/dx）

d²y/dx² + 3 dy/dx + 2y = sin x → Second-order (highest derivative is d²y/dx²)

d²y/dx² + 3 dy/dx + 2y = sin x → 二阶（最高导数为 d²y/dx²）

It is important to note that the order is determined solely by the highest derivative, regardless of how many lower-order derivatives appear. An equation containing dy/dx and y, but no higher derivatives, is always first-order.

需要注意，阶仅由最高阶导数决定，无论方程中出现多少个低阶导数。一个包含 dy/dx 和 y 但不含更高阶导数的方程始终是一阶的。

3. First-Order Differential Equations | 一阶微分方程

A first-order differential equation involves only the first derivative dy/dx, the dependent variable y, and the independent variable x. The general form is often written as dy/dx = f(x, y), meaning that the gradient at any point (x, y) is given by a function of both coordinates.

一阶微分方程只涉及一阶导数 dy/dx、因变量 y 和自变量 x。其一般形式通常写成 dy/dx = f(x, y)，意味着在任意点 (x, y) 处的梯度由关于这两个坐标的函数给出。

The simplest first-order equations can be solved by direct integration:

最简单的一阶方程可以通过直接积分求解：

dy/dx = 6x² → y = 2x³ + C

Here C is an arbitrary constant of integration. Because one arbitrary constant appears in the general solution, a first-order equation requires one initial condition to determine a unique particular solution.

这里 C 是任意积分常数。由于通解中出现一个任意常数，一阶方程需要有一个初始条件来确定唯一的特解。

4. Second-Order Differential Equations | 二阶微分方程

A second-order differential equation contains the second derivative d²y/dx² as its highest derivative. Such equations are common in physics, particularly in describing simple harmonic motion, where acceleration (the second derivative of displacement) is proportional to displacement.

二阶微分方程以二阶导数 d²y/dx² 作为其最高阶导数。这类方程在物理学中很常见，特别是在描述简谐运动时，加速度（位移的二阶导数）与位移成正比。

The general second-order linear equation with constant coefficients has the form:

常系数二阶线性方程的一般形式为：

a d²y/dx² + b dy/dx + cy = f(x)

where a, b and c are constants. If a ≠ 0, the equation is genuinely second-order. The general solution of a second-order equation typically contains two arbitrary constants, so two initial conditions are needed to find a particular solution.

其中 a、b 和 c 为常数。若 a ≠ 0，则该方程是真正的二阶方程。二阶方程的通解通常包含两个任意常数，因此需要两个初始条件才能确定特解。

5. Linearity: Definition | 线性：定义

A differential equation is linear if the dependent variable y and all of its derivatives appear only to the first power, are not multiplied together, and are not arguments of other functions such as sin(y) or eʸ. In other words, the equation must be a linear combination of y, y′, y″, and so on, with coefficients that depend only on x.

如果因变量 y 及其所有导数都只以一次幂出现、彼此不相乘、也不是其他函数的自变量（如 sin(y) 或 eʸ），则微分方程是线性的。换言之，方程必须是 y、y′、y″ 等的线性组合，且系数仅依赖于 x。

For example, the following equations are linear because each term involving y is of degree one:

例如，以下方程是线性的，因为每个涉及 y 的项都是一次的：

3 dy/dx + 2y = x²

d²y/dx² + 5 dy/dx + 6y = 0

In both cases, each derivative term is raised to the first power and appears alone. The right-hand side may be any function of x, but it must not contain y.

在上述两个例子中，每个导数项都是一次幂并且单独出现。右侧可以是任何关于 x 的函数，但不能含有 y。

6. Non-Linear Differential Equations | 非线性微分方程

If a differential equation contains any term where y or its derivatives are raised to a power greater than one, multiplied together, or placed inside a function such as sin(y) or ln(y), it is called non-linear. Non-linear equations are generally far more difficult to solve exactly, and many have no elementary closed-form solution.

如果微分方程中包含任何 y 或其导数幂次大于一、彼此相乘、或置于 sin(y)、ln(y) 等函数内部的情况，则称为非线性方程。非线性方程通常很难精确求解，许多非线性方程甚至没有初等闭式解。

Examples of non-linear equations include:

非线性方程的例子包括：

(dy/dx)² + y = 0 — the derivative is squared, so non-linear
(dy/dx)² + y = 0 — 导数被平方，故非线性
d²y/dx² + sin(y) = 0 — y appears inside sin, so non-linear
d²y/dx² + sin(y) = 0 — y 出现在 sin 函数内，故非线性
dy/dx = xy² — y is squared, so non-linear
dy/dx = xy² — y 被平方，故非线性

For the AQA A-Level specification, you are only required to solve linear differential equations, but you must be able to recognise non-linear equations and explain why they are non-linear.

在 AQA A-Level 考纲中，只要求你求解线性微分方程，但你必须能够识别非线性方程并解释其非线性的原因。

7. Homogeneous and Non-Homogeneous Linear Equations | 齐次与非齐次线性方程

Within the class of linear differential equations, there is a further important distinction between homogeneous and non-homogeneous equations. A linear differential equation is homogeneous if the right-hand side is zero; that is, every term contains either y or one of its derivatives.

在线性微分方程中，还有一个重要的区分：齐次与非齐次。如果线性微分方程的右侧为零，即每一项都含有 y 或其导数，则该方程是齐次的。

d²y/dx² + 4y = 0 → Homogeneous

d²y/dx² + 4y = 0 → 齐次

d²y/dx² + 4y = e²ˣ → Non-homogeneous

d²y/dx² + 4y = e²ˣ → 非齐次

A non-homogeneous equation has a non-zero term f(x) on the right-hand side that does not involve y. The solution to a non-homogeneous equation is the sum of the complementary function (the general solution of the corresponding homogeneous equation) and a particular integral (one solution to the full non-homogeneous equation).

非齐次方程的右侧有一个不含 y 的非零项 f(x)。非齐次方程的解是补充函数（对应齐次方程的通解）与特积分（整个非齐次方程的一个解）之和。

8. Verification of Solutions | 解的验证

To check whether a given function is a solution of a differential equation, we substitute the function and its derivatives into the equation and verify that the left-hand side equals the right-hand side for all x in the relevant domain.

要验证一个给定函数是否为微分方程的解，我们将该函数及其导数代入方程，并检查在相关定义域内左侧是否恒等于右侧。

Consider the differential equation dy/dx = 2y. Suppose we wish to test whether y = e²ˣ is a solution. Since dy/dx = 2e²ˣ and 2y = 2e²ˣ, the equation is satisfied, so y = e²ˣ is indeed a solution.

考虑微分方程 dy/dx = 2y。假设我们要检验 y = e²ˣ 是否为其解。由于 dy/dx = 2e²ˣ 且 2y = 2e²ˣ，方程成立，因此 y = e²ˣ 确实是一个解。

In fact, the general solution is y = Ce²ˣ for any constant C. This demonstrates an important property of linear homogeneous equations: if y₁ and y₂ are solutions, then any linear combination C₁y₁ + C₂y₂ is also a solution. This principle is known as the superposition principle.

实际上，通解是 y = Ce²ˣ（C 为任意常数）。这表明了线性齐次方程的一个重要性质：若 y₁ 和 y₂ 是解，则它们的任意线性组合 C₁y₁ + C₂y₂ 也是解。这一原理称为叠加原理。

9. Initial Conditions and Boundary Conditions | 初始条件与边界条件

An initial condition specifies the value of the solution (and possibly its derivatives) at a single point, typically x = 0. For a first-order equation, one initial condition such as y(0) = 3 is enough to determine the arbitrary constant. For a second-order equation, two conditions are needed.

初始条件指定了解在单个点（通常为 x = 0）处的值（以及可能的导数值）。对于一阶方程，一个初始条件（如 y(0) = 3）足以确定任意常数；对于二阶方程，则需要两个条件。

For example, the first-order equation dy/dx = 2y has the general solution y = Ce²ˣ. Applying y(0) = 3 gives 3 = C · e⁰ = C, so the particular solution is y = 3e²ˣ.

例如，一阶方程 dy/dx = 2y 的通解为 y = Ce²ˣ。代入 y(0) = 3 得 3 = C · e⁰ = C，因此特解为 y = 3e²ˣ。

Boundary conditions, by contrast, specify the value of the solution at two different points. They arise in problems involving the shape of a string or beam, but at A-Level they are less commonly examined than initial-value problems.

相比之下，边界条件在两个不同点处指定解的值。它们出现在涉及弦或梁的形状问题中，但在 A-Level 中，边界条件问题的考查频率低于初值问题。

10. Forming Differential Equations from Real-World Problems | 从实际问题建立微分方程

A key skill in the AQA specification is translating a written statement into a differential equation. The phrase “rate of change” always indicates a derivative, and the word “proportional” indicates multiplication by a constant.

AQA 考纲中的一项关键技能是将文字表述转化为微分方程。”变化率”一词总是表示导数，而”成正比”表示乘以一个常数。

Suppose a radioactive substance decays at a rate proportional to the amount N of substance present. The rate of change of N with respect to time t is dN/dt. Since the substance is decaying, this rate is negative:

假设某放射性物质的衰变速率与现存物质的量 N 成正比。N 关于时间 t 的变化率是 dN/dt。由于物质在衰变，该速率为负：

dN/dt = -kN, where k > 0 is the decay constant

dN/dt = -kN，其中 k > 0 为衰变常数

This is a first-order, linear, homogeneous differential equation. Its solution is an exponential decay function N = N₀e⁻ᵏᵗ, where N₀ is the initial amount.

这是一个一阶线性齐次微分方程。其解为指数衰减函数 N = N₀e⁻ᵏᵗ，其中 N₀ 是初始量。

11. Classifying Equations: A Systematic Approach | 方程分类：系统化方法

When faced with an unfamiliar differential equation, A-Level students should follow a systematic three-step classification procedure. First, identify the highest derivative to determine the order. Second, check whether y and all its derivatives appear to the first power only, without products or compositions. Third, check whether the right-hand side is zero to determine homogeneity.

面对一个不熟悉的微分方程时，A-Level 学生应遵循系统的三步分类法。首先，确定最高阶导数以判断阶数；其次，检查 y 及其所有导数是否只以一次幂出现，且没有乘积或复合；第三，检查右侧是否为零以判断齐次性。

Consider the equation d²y/dx² + 3(dy/dx)² + y = 0. The highest derivative is d²y/dx², so it is second-order. However, the term (dy/dx)² involves the first derivative squared, so the equation is non-linear. Because the right-hand side is zero, it is homogeneous — but homogeneity only applies to linear equations, so we simply call it a non-linear second-order equation.

考虑方程 d²y/dx² + 3(dy/dx)² + y = 0。最高阶导数为 d²y/dx²，因此是二阶的。然而，项 (dy/dx)² 涉及一阶导数的平方，因此该方程是非线性的。由于右侧为零，它是齐次的——但齐次性仅适用于线性方程，因此我们直接称之为非线性二阶方程。

This classification matters because it determines which solution method is available. The AQA specification expects you to solve first-order linear equations using integrating factors, and second-order linear equations with constant coefficients using auxiliary equations.

这种分类很重要，因为它决定了可用的求解方法。AQA 考纲期望你使用积分因子求解一阶线性方程，并使用辅助方程求解常系数二阶线性方程。

12. Common Pitfalls and Exam Tips | 常见错误与考试技巧

Students frequently confuse order with degree. The order of the highest derivative is a classification of the equation’s structure, whereas the degree refers to the power to which that highest derivative is raised. In A-Level AQA, the most important check is simply the order and linearity.

学生经常混淆阶与次。最高导数的阶是对方程结构的分类，而次指的是该最高导数被提升的幂次。在 A-Level AQA 中，最重要的检查就是阶与线性。

Another common error is failing to spot non-linearity when a term like y dy/dx appears. Although each factor appears singly, the product of y and dy/dx means the equation is non-linear. Similarly, expressions like sin(y) or eʸ immediately indicate a non-linear equation.

另一个常见错误是当出现 y dy/dx 这样的项时未能识别非线性。虽然每个因子单独出现，但 y 与 dy/dx 的乘积意味着方程是非线性的。同样，sin(y) 或 eʸ 这样的表达式也立即表示方程是非线性的。

Finally, always write down the order and linearity status explicitly in examinations when asked. Examiners award marks for correct classification, and stating “this is a first-order linear differential equation” at the start of a solution shows clear understanding.

最后，考试中被问及时，务必明确写出阶数和线性状态。考官会对正确的分类给分，在解题开头写出”这是一个一阶线性微分方程”能表明你理解清晰。

Remember that for a linear equation, the power of x on the right-hand side is irrelevant to classification; it may be x², eˣ, or sin x, and the equation remains linear. Only the way y and its derivatives appear matters.

请记住，对于线性方程，右侧 x 的幂次与分类无关；它可以是 x²、eˣ 或 sin x，方程仍然是线性的。只有 y 及其导数的出现方式才是关键。

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