📚 The concept of a differential equation: order and linearity | 微分方程的概念：阶与线性
Differential equations are equations that relate an unknown function to its derivatives. They appear throughout AQA A-level Mathematics when modelling rates of change, population growth, cooling and motion. Before learning solution methods, you must be able to classify a differential equation by its order and decide whether it is linear. These two concepts determine which techniques can be applied and how many conditions are needed for a unique solution.
微分方程是联系未知函数及其导数的方程。在 AQA A-level 数学中，它们广泛用于建立变化率、人口增长、冷却和运动等模型。在学习求解方法之前，你必须能够按阶对微分方程进行分类，并判断它是否为线性。这两个概念决定了可以应用哪些技巧，以及获得唯一解需要多少个条件。
1. What is a differential equation? | 什么是微分方程？
lution is not a single number but a function y = x² + C, where C is an arbitrary constant.
微分方程是包含未知函数（通常记作 y 或 f(x)）及其一个或多个导数的方程。例如，dy/dx = 2x 表示 y 对 x 的导数等于 2x。它的解不是一个数，而是一个函数 y = x² + C，其中 C 为任意常数。
Because derivatives measure rates of change, differential equations naturally describe dynamic processes. In AQA questions you may be asked to identify the dependent and independent variables, or to verify that a given function satisfies a differential equation.
由于导数测量变化率，微分方程自然描述动态过程。在 AQA 考题中，你可能需要识别因变量和自变量，或验证某个给定函数是否满足微分方程。
2. Notation and terminology | 记号与术语
Common notations for the first derivative include dy/dx, y' and f'(x). The second derivative may be written d²y/dx² or y''. In a differential equation, x is usually the independent variable and y is the dependent variable.
一阶导数的常用记号包括 dy/dx、y' 和 f'(x)。二阶导数可以写作 d²y/dx² 或 y''。在微分方程中，x 通常是自变量，y 是因变量。
x² + C. If we also require y(0) = 3, then C = 3 and the particular solution is y = x² + 3.
通解包含任意常数；特解满足附加条件并确定这些常数。例如，dy/dx = 2x 的通解是 y = x² + C。如果还要求 y(0) = 3，则 C = 3，特解为 y = x² + 3。
3. Order of a differential equation | 微分方程的阶
order of the highest derivative that appears in the equation. For instance, dy/dx + 2y = eˣ . The equation d²y/dx² + 3dy/dx + 2y = 0 is second-order because the term d²y/dx² appears.
微分方程的阶是方程中出现的最高阶导数的阶数。例如，dy/dx + 2y = eˣ 是一阶方程，因为最高导数是 dy/dx。方程 d²y/dx² + 3dy/dx + 2y = 0 是二阶方程，因为出现了 d²y/dx²。
An equation containing d³y/dx³ would be third-order, even if lower derivatives are also present. The order is not affected by powers of x, by constants, or by the arrangement of terms. You should locate the highest derivative first and ignore everything else when stating the order.
含有 d³y/dx³ 的方程是三阶方程，即使其中也含有较低阶导数。阶数不受 x 的幂、常数或项的安排影响。在说明阶数时，应首先找出最高阶导数，并忽略其他一切。
Order = highest derivative order present
4. Linearity: first idea | 线性：初步概念
A differential equation is linear if the dependent variable y and all its derivatives appear only to the first power, are not multiplied together, and are not inside nonlinear functions such as sin y, eʸ or ln y. Coefficients may depend on the independent variable x.
如果因变量 y 及其所有导数只以一次幂出现，彼此不相乘，也不出现在 sin y、eʸ 或 ln y 等非线性函数中，则该微分方程是线性的。系数可以依赖于自变量 x。
near. The condition for linearity applies only to y and its derivatives, never to x alone.
例如，x² dy/dx + 3y = cos x 是线性方程，而 y dy/dx + y = 0 和 (dy/dx)² + y = 0 是非线性方程。线性条件只适用于 y 及其导数，绝不单独适用于 x。
5. Linear vs nonlinear examples | 线性与非线性的例子
Classifying an equation correctly requires scanning each term. Compare the following examples:
正确分类方程需要检查每一项。比较以下例子：
| Equation | Linear? | Reason |
|---|---|---|
| d²y/dx² + 5dy/dx + 6y = 0 | Yes | y, y' and y'' appear to the first power only |
| dy/dx + (cos x)y = x² | Yes | cos x is a coefficient of x, not a function of y |
| y dy/dx + y = 0 | No | the product y(dy/dx) breaks linearity |
| (dy/dx)² + y =
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