📚 Differential Equations: Standard Forms and Solution Methods | 微分方程：标准形式与解法
A differential equation is an equation that involves a function and its derivatives. In A-level mathematics, solving first-order differential equations is a core skill that connects calculus with real-world modelling. This article explores the standard forms you will meet in the AQA specification, the techniques for solving them, and the common traps to avoid.
微分方程是包含函数及其导数的方程。在 A-level 数学中，解一阶微分方程是连接微积分与实际问题建模的核心技能。本文将探讨 AQA 考纲中要求掌握的标准形式、求解方法以及常见的易错点。
1. What Is a Differential Equation? | 什么是微分方程？
A differential equation describes a relationship between a variable, its derivatives, and possibly the independent variable. For example, the equation dy/dx = 2x states that the gradient of a curve at every point equals twice the x-coordinate. The general solution contains an arbitrary constant, which can be found from an initial condition or boundary condition.
微分方程描述了一个变量、其导数以及可能还有自变量之间的关系。例如，方程 dy/dx = 2x 表示曲线在每个点的斜率等于该点 x 坐标的两倍。通解会包含一个任意常数，该常数可通过初始条件或边界条件来确定。
2. Standard Forms of First-Order Equations | 一阶方程的标准形式
For the AQA course, the most important first-order differential equations can be written in one of two standard forms. The first is the separable form: dy/dx = f(x)g(y). The second is the linear form: dy/dx + P(x)y = Q(x). Recognising which form an equation takes is the key to choosing the correct solution method.
在 AQA 课程中，最重要的一阶微分方程可以写成两种标准形式之一。第一种是可分离形式：dy/dx = f(x)g(y)。第二种是线性形式：dy/dx + P(x)y = Q(x)。识别一个方程属于哪种形式，是选择正确解法的关键。
- Separable form: dy/dx = f(x)g(y) – variables can be separated to opposite sides.
- Linear form: dy/dx + P(x)y = Q(x) – solved using an integrating factor.
- 可分离形式：dy/dx = f(x)g(y) —— 变量可分离到等号两边。
- 线性形式：dy/dx + P(x)y = Q(x) —— 使用积分因子求解。
3. Separable Differential Equations | 可分离变量的微分方程
To solve a separable equation, we rearrange the expression so that all terms involving y are on one side and all terms involving x are on the other side. Then we integrate both sides. Algebraically, we treat dy/dx as a ratio for this manipulation, which is accepted at A-level.
要解可分离方程，我们重新整理表达式，使所有含 y 的项在一侧，所有含 x 的项在另一侧。然后两边积分。在代数上，我们为了这种操作将 dy/dx 视为一个比，这在 A-level 中是允许的。
dy/dx = f(x)g(y) → ∫ 1/g(y) dy = ∫ f(x) dx
Once the integration is completed, the result is an implicit relation between x and y. If required, this can be rearranged to give y explicitly in terms of x.
一旦完成积分，结果就是 x 与 y 之间的隐式关系。如果需要，可以重新整理以得到 y 关于 x 的显式表达式。
4. Worked Example 1: Separable Form | 实例 1：可分离形式
Solve dy/dx = x^2 / y^2, given that y(0) = 1.
求解 dy/dx = x² / y²，已知 y(0) = 1。
First separate the variables: y^2 dy = x^2 dx.
首先分离变量：y² dy = x² dx。
Integrate both sides: ∫ y^2 dy = ∫ x^2 dx, so y^3/3 = x^3/3 + C.
两边积分：∫ y² dy = ∫ x² dx，所以 y³/3 = x³/3 + C。
Multiply by 3: y^3 = x^3 + 3C. Let the new constant be D = 3C, so y^3 = x^3 + D.
两边乘以 3：y³ = x³ + 3C。令新常数 D = 3C，所以 y³ = x³ + D。
Use y(0) = 1: 1 = 0 + D ⇒ D = 1. Hence y = (x^3 + 1)^(1/3).
利用 y(0) = 1：1 = 0 + D ⇒ D = 1。因此 y = (x³ + 1)^(1/3)。
5. First-Order Linear Equations | 一阶线性方程
A first-order linear differential equation can be written in the form dy/dx + P(x)y = Q(x). The left-hand side resembles the derivative of a product, but the two terms are not exactly a product rule unless we multiply by a suitable function called the integrating factor.
一阶线性微分方程可以写成 dy/dx + P(x)y = Q(x) 的形式。左侧看起来像乘积的导数，但除非我们乘以一个适当的函数，即所谓积分因子，否则这两项并不完全是乘积法则。
Integrating factor = e^(∫ P(x) dx)
The reason for this factor is that when we multiply the whole equation by it, the left side becomes the derivative of (y × integrating factor).
使用这个因子的原因是，当我们把整个方程乘以它时，左侧就变成了 (y × 积分因子) 的导数。
6. Deriving the Integrating Factor | 推导积分因子
Suppose we multiply dy/dx + P(x)y = Q(x) by an unknown function I(x): I(x)dy/dx + I(x)P(x)y = I(x)Q(x).
假设我们将 dy/dx + P(x)y = Q(x) 乘以一个未知函数 I(x)：I(x)dy/dx + I(x)P(x)y = I(x)Q(x)。
We want the left side to equal (I(x)y)’ = I(x)dy/dx + I'(x)y. Comparing the coefficients of y, we need I'(x) = I(x)P(x).
我们希望左侧等于 (I(x)y)’ = I(x)dy/dx + I'(x)y。比较 y 的系数，我们需要 I'(x) = I(x)P(x)。
This is itself a separable equation: dI/dx = P(x)I. Separating gives (1/I) dI = P(x) dx, and integrating produces ln|I| = ∫ P(x) dx. Therefore I = e^(∫ P(x) dx).
这本身就是一个可分离方程：dI/dx = P(x)I。分离变量得到 (1/I) dI = P(x) dx，积分得 ln|I| = ∫ P(x) dx。因此 I = e^(∫ P(x) dx)。
7. Worked Example 2: Linear Form | 实例 2：线性形式
Solve dy/dx + 2y = e^(3x), with y(0) = 1.
求解 dy/dx + 2y = e^(3x)，且 y(0) = 1。
Here P(x) = 2, Q(x) = e^(3x). The integrating factor is e^(∫ 2 dx) = e^(2x).
这里 P(x) = 2，Q(x) = e^(3x)。积分因子为 e^(∫ 2 dx) = e^(2x)。
Multiply the equation by e^(2x): e^(2x)dy/dx + 2e^(2x)y = e^(5x).
将方程乘以 e^(2x)：e^(2x)dy/dx + 2e^(2x)y = e^(5x)。
The left side is (y e^(2x))’. Integrate both sides: y e^(2x) = ∫ e^(5x) dx = (1/5)e^(5x) + C.
左侧是 (y e^(2x))’。两边积分：y e^(2x) = ∫ e^(5x) dx = (1/5)e^(5x) + C。
So y = (1/5)e^(3x) + C e^(-2x). Applying y(0) = 1 gives 1 = 1/5 + C ⇒ C = 4/5.
所以 y = (1/5)e^(3x) + C e^(-2x)。应用 y(0) = 1 得 1 = 1/5 + C ⇒ C = 4/5。
Final solution: y = (1/5)e^(3x) + (4/5)e^(-2x).
最终解：y = (1/5)e^(3x) + (4/5)e^(-2x)。
8. Initial Conditions and Particular Solutions | 初始条件与特解
The general solution of a first-order differential equation always contains one arbitrary constant. An initial condition is an extra piece of information that fixes this constant, producing a particular solution. This condition often takes the form y(x₀) = y₀.
一阶微分方程的通解总是包含一个任意常数。初始条件是确定该常数的额外信息，从而产生特解。该条件通常采用 y(x₀) = y₀ 的形式。
When applying an initial condition, substitute the x- and y-values into the general solution and solve for the constant. Be careful: if the general solution is implicit, you might need to solve an equation involving the constant.
在应用初始条件时，将 x 和 y 的值代入通解并求解常数。注意：如果通解是隐式形式，你可能需要解一个包含常数的方程。
9. Application: Exponential Growth and Decay | 应用：指数增长与衰退
A classic model is dy/dt = k y, where y is a quantity and t is time. This is separable and leads to y = A e^(kt). If k > 0, it represents exponential growth; if k < 0, it represents exponential decay.
经典模型是 dy/dt = k y，其中 y 是某个量，t 是时间。这是可分离的，得到 y = A e^(kt)。若 k > 0，表示指数增长；若 k < 0，表示指数衰退。
dy/dt = k y → y = A e^(kt)
This equation appears in radioactivity, continuous compound interest, and population growth. The constant A is the initial amount at t = 0.
这个方程出现在放射性、连续复利和人口增长等问题中。常数 A 是 t = 0 时的初始量。
10. Application: Newton’s Law of Cooling | 应用：牛顿冷却定律
Newton’s law of cooling states that the rate of change of temperature T of an object is proportional to the difference between T and the surrounding temperature Tₛ. That is, dT/dt = -k(T – Tₛ).
牛顿冷却定律指出，物体温度 T 的变化率与 T 和周围环境温度 Tₛ 之差成正比。即 dT/dt = -k(T – Tₛ)。
This equation is separable: ∫ 1/(T – Tₛ) dT = ∫ -k dt, giving ln|T – Tₛ| = -kt + C, and T = Tₛ + A e^(-kt).
该方程是可分离的：∫ 1/(T – Tₛ) dT = ∫ -k dt，得到 ln|T – Tₛ| = -kt + C，进而 T = Tₛ + A e^(-kt)。
These applications show why the standard forms are so valuable – they turn a physical situation into mathematics you can solve with confidence.
这些应用显示了标准形式的价值——它们将物理情景转化为你能自信求解的数学。
11. Common Pitfalls | 常见易错点
Here are the typical mistakes students make when solving differential equations in A-level exams.
以下是 A-level 考试中学生解微分方程时犯的典型错误。
- Failing to separate variables properly – always check that you have factored f(x)g(y) correctly.
- Forgetting the constant of integration when integrating both sides.
- Misidentifying P(x) in the linear form, especially when there is a coefficient before dy/dx.
- Forgetting to multiply Q(x) by the integrating factor when integrating.
- Dropping modulus signs when integrating 1/y, although this is often acceptable if y > 0.
- 未能正确分离变量——始终检查是否已正确分解 f(x)g(y)。
- 积分时忘记积分常数。
- 在 dy/dx 前有系数时错误识别线性形式中的 P(x)。
- 积分时忘记将 Q(x) 乘以积分因子。
- 积分 1/y 时丢弃绝对值符号，虽然当 y > 0 时通常可以接受。
12. Summary and Revision Tips | 总结与复习建议
To solve any first-order differential equation on the AQA specification, first decide whether it is separable or linear. Try to rewrite dy/dx in a factored form first; if that is possible, use separation. Otherwise, check if it can be cast as dy/dx + P(x)y = Q(x).
要解 AQA 考纲中的任何一阶微分方程，首先要判断它是可分离的还是线性的。尝试先将 dy/dx 写成因式分解的形式；如果可能，就用分离变量法。否则，检查是否能将其转化为 dy/dx + P(x)y = Q(x) 的形式。
Practise with past exam questions, and always verify your solution by differentiating it and substituting back into the original equation. This catches most algebraic errors.
用往年真题进行练习，并始终通过微分你的解并代回原方程来验证它。这能发现大多数代数错误。
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