Differential Equations Fundamentals and Solution Strategies | 微分方程基础与求解策略

📚 Differential Equations Fundamentals and Solution Strategies | 微分方程基础与求解策略

A differential equation is a mathematical equation that relates a function with its derivatives. In A-Level mathematics, differential equations are a core topic in both pure mathematics and applied contexts, forming the bridge between algebraic manipulation and real-world modeling.

微分方程是联系函数与其导数的数学方程。在A-Level数学中，微分方程是纯数学与应用情境中的核心考点，是连接代数运算与现实建模的桥梁。

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

A differential equation involves an unknown function and one or more of its derivatives. The order of a differential equation is determined by the highest derivative present. For example, dy/dx = 2x is a first-order equation, while d²y/dx² + y = 0 is second-order.

微分方程包含一个未知函数及其一个或多个导数。阶数由方程中最高阶导数决定。例如，dy/dx = 2x 是一阶方程，而 d²y/dx² + y = 0 是二阶方程。

The degree of a differential equation is the power of the highest-order derivative after the equation has been made rational and integral in all derivatives. Most A-Level questions involve first-degree (linear) differential equations.

次数是在方程化为有理整式后最高阶导数的幂次。大多数A-Level题目涉及一次（线性）微分方程。

A general solution contains arbitrary constants; a particular solution is obtained by substituting initial conditions or boundary conditions to find those constants.

通解含有任意常数；通过代入初始条件或边界条件求得具体常数后，得到特解。

2. Classifying Differential Equations | 微分方程的分类

In the A-Level syllabus, differential equations are classified primarily in two ways: by order (first-order or second-order) and by linearity (linear or non-linear). A linear differential equation has the unknown function and its derivatives appearing only to the first power, with no products of these terms.

在A-Level大纲中，微分方程主要从两个角度分类：按阶数（一阶或二阶）和按线性（线性或非线性）。线性微分方程中未知函数及其导数只以一次幂出现，且不包含它们的乘积项。

A first-order linear differential equation can be written in the general form:

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

where P(x) and Q(x) are functions of x only. Similarly, a second-order linear differential equation has the form a·d²y/dx² + b·dy/dx + c·y = f(x), where a, b, c are constants.

一阶线性微分方程可写成一般形式：

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

其中 P(x) 和 Q(x) 仅是 x 的函数。类似地，二阶线性微分方程具有形式 a·d²y/dx² + b·dy/dx + c·y = f(x)，其中 a、b、c 为常数。

3. Method of Separation of Variables | 分离变量法

The separation of variables method is applicable when a first-order differential equation can be rearranged so that all terms involving y (and dy) are on one side, and all terms involving x (and dx) are on the other side. This technique is a cornerstone of the A-Level 微分方程 unit.

分离变量法适用于可以整理为一边只含 y（和 dy）、另一边只含 x（和 dx）的一阶微分方程。这一技巧是A-Level微分方程单元的基石。

Step 1: Rearrange the equation into the form f(y) dy = g(x) dx.

步骤一：将方程整理为 f(y) dy = g(x) dx 的形式。

Step 2: Integrate both sides: ∫ f(y) dy = ∫ g(x) dx.

步骤二：两边同时积分：∫ f(y) dy = ∫ g(x) dx。

Step 3: Add the constant of integration C and solve for y explicitly if possible.

步骤三：加上积分常数 C，尽可能显式解出 y。

Consider the worked example: dy/dx = 3x²y. Separating variables gives (1/y) dy = 3x² dx. Integrating both sides yields ln|y| = x³ + C, hence y = Ae^(x³), where A = ±e^C.

看一个经典例题：dy/dx = 3x²y。分离变量得到 (1/y) dy = 3x² dx。两边积分得 ln|y| = x³ + C，因此 y = Ae^(x³)，其中 A = ±e^C。

Key examination tip: always include the absolute value in ln|y|, and rewrite ±e^C as a single constant A before finalizing the answer.

关键考试提示：ln 中始终保留绝对值 |y|，并在最终答案前将 ±e^C 合并为单一常数 A。

4. Integrating Factor Method | 积分因子法

When a first-order differential equation of the form dy/dx + P(x)y = Q(x) cannot be separated, the integrating factor method is the standard approach. The integrating factor is defined as:

当形如 dy/dx + P(x)y = Q(x) 的一阶微分方程无法分离变量时，积分因子法成为标准解法。积分因子定义为：

I = e^(∫ P(x) dx)

Multiplying both sides of the differential equation by I converts the left-hand side into the derivative of I·y:

将微分方程两边同乘以 I，可将左边化为 I·y 的导数：

d/dx (I·y) = I·Q(x)

Then integrate both sides: I·y = ∫ I·Q(x) dx + C, and finally divide by I to obtain y.

然后两边积分：I·y = ∫ I·Q(x) dx + C，最后除以 I 得到 y。

Let us examine an example: dy/dx + 2y/x = x³. Here P(x) = 2/x. The integrating factor is I = e^(∫ 2/x dx) = e^(2 ln|x|) = x². Multiplying through gives x² · dy/dx + 2xy = x⁵, which simplifies to d/dx (x²y) = x⁵. Integrating: x²y = x⁶/6 + C, so y = x⁴/6 + C/x².

我们来看一个例子：dy/dx + 2y/x = x³。这里 P(x) = 2/x。积分因子为 I = e^(∫ 2/x dx) = e^(2 ln|x|) = x²。两边同乘得 x² · dy/dx + 2xy = x⁵，即 d/dx (x²y) = x⁵。积分：x²y = x⁶/6 + C，所以 y = x⁴/6 + C/x²。

5. Second-Order Homogeneous Equations | 二阶齐次方程

A second-order homogeneous linear differential equation with constant coefficients has the form a·d²y/dx² + b·dy/dx + c·y = 0. To solve it, we assume a solution of the form y = e^(λx), which leads to the auxiliary equation:

常系数二阶齐次线性微分方程具有形式 a·d²y/dx² + b·dy/dx + c·y = 0。求解时假设解具有形式 y = e^(λx)，从而得到辅助方程：

aλ² + bλ + c = 0

There are three cases to consider based on the discriminant Δ = b² − 4ac:

根据判别式 Δ = b² − 4ac 分为三种情况：

If Δ > 0: two distinct real roots λ₁, λ₂, giving the general solution y = Ae^(λ₁x) + Be^(λ₂x).

若 Δ > 0：两个不等实根 λ₁、λ₂，通解为 y = Ae^(λ₁x) + Be^(λ₂x)。
If Δ = 0: one repeated real root λ, giving y = (A + Bx)e^(λx).

若 Δ = 0：一个二重实根 λ，通解为 y = (A + Bx)e^(λx)。
If Δ < 0: complex conjugate roots λ = p ± qi, giving y = e^(px)(A cos(qx) + B sin(qx)).

若 Δ < 0：共轭复根 λ = p ± qi，通解为 y = e^(px)(A cos(qx) + B sin(qx))。

For example, the equation d²y/dx² − 5·dy/dx + 6y = 0 has auxiliary equation λ² − 5λ + 6 = 0 = (λ − 2)(λ − 3). Thus the general solution is y = Ae^(2x) + Be^(3x).

例如，方程 d²y/dx² − 5·dy/dx + 6y = 0 的辅助方程为 λ² − 5λ + 6 = 0 = (λ − 2)(λ − 3)。因此通解为 y = Ae^(2x) + Be^(3x)。

6. Second-Order Non-Homogeneous Equations | 二阶非齐次方程

A non-homogeneous equation has the form a·d²y/dx² + b·dy/dx + c·y = f(x). The general solution is the sum of the complementary function (CF, the solution of the corresponding homogeneous equation) and a particular integral (PI, any one solution of the full equation).

非齐次方程具有形式 a·d²y/dx² + b·dy/dx + c·y = f(x)。通解为互补函数（CF，对应齐次方程的解）与特解（PI，原方程的任意一个解）之和。

The table below summarises the standard trial functions for choosing a PI:

下表总结了选择特解时的标准试探函数：

f(x) 的形式	试探特解形式
多项式（如 x²）	同次数多项式（如 ax² + bx + c）
指数型（如 e^(kx)）	Ae^(kx)（若 k 是辅助方程的根，则需乘 x）
三角函数（如 sin(kx) 或 cos(kx)）	A sin(kx) + B cos(kx)

Consider d²y/dx² + y = 3 sin(2x). The CF is y = A cos(x) + B sin(x). For the PI, try y = p sin(2x) + q cos(2x). Substitution gives (−4p sin(2x) − 4q cos(2x)) + (p sin(2x) + q cos(2x)) = 3 sin(2x). Therefore −3p = 3 and −3q = 0, so p = −1, q = 0. The PI is y = −sin(2x), and the general solution is y = A cos(x) + B sin(x) − sin(2x).

考虑 d²y/dx² + y = 3 sin(2x)。CF 为 y = A cos(x) + B sin(x)。对特解尝试 y = p sin(2x) + q cos(2x)。代入得 (−4p sin(2x) − 4q cos(2x)) + (p sin(2x) + q cos(2x)) = 3 sin(2x)。因此 −3p = 3、−3q = 0，所以 p = −1、q = 0。特解为 y = −sin(2x)，通解为 y = A cos(x) + B sin(x) − sin(2x)。

Special caution: if the trial function duplicates a term in the CF, multiply the trial by x (or x² in resonance cases) to avoid linear dependence.

特别注意：若试探解与CF中的某一项重复，需将试探解乘以 x（共振情形下乘 x²）以避免线性相关。

7. Initial Value Problems and Boundary Conditions | 初值问题与边界条件

An initial value problem specifies conditions at a single point, typically y(x₀) = y₀ and y′(x₀) = y₁. A boundary value problem specifies conditions at two different points. Both are resolved by substituting the conditions into the general solution to find the arbitrary constants.

初值问题在单个点处给出条件，通常为 y(x₀) = y₀ 和 y′(x₀) = y₁。边值问题在两个不同点处给出条件。两者的求解方法都是将条件代入通解以确定任意常数。

Worked example: Solve dy/dx = 6x² − 4x with y(1) = 5. Integrating gives y = 2x³ − 2x² + C. Substituting x = 1, y = 5 gives 5 = 2 − 2 + C, so C = 5. Therefore y = 2x³ − 2x² + 5.

例题：求解 dy/dx = 6x² − 4x，且 y(1) = 5。积分得 y = 2x³ − 2x² + C。代入 x = 1、y = 5 得 5 = 2 − 2 + C，即 C = 5。因此 y = 2x³ − 2x² + 5。

For second-order initial value problems, both y and dy/dx at x₀ are used to determine the two constants A and B simultaneously.

对于二阶初值问题，利用 x₀ 处的 y 和 dy/dx 两个条件联立求出常数 A 和 B。

8. Applications in Real-World Modelling | 微分方程的实际建模应用

Differential equations are powerful tools for modelling real-world phenomena. In A-Level examinations, common applications include population growth (unrestricted and restricted), radioactive decay, Newton’s law of cooling, and simple harmonic motion.

微分方程是模拟现实世界现象的有力工具。在A-Level考试中，常见应用包括人口增长（无限制与有限制）、放射性衰变、牛顿冷却定律和简谐运动。

Exponential growth/decay: dN/dt = kN, where k is the growth (k > 0) or decay (k < 0) constant. The solution is N = N₀e^(kt), where N₀ is the initial population.

指数增长/衰减：dN/dt = kN，其中 k 为增长（k > 0）或衰减（k < 0）常数。解为 N = N₀e^(kt)，其中 N₀ 为初始数量。

Restricted growth (logistic model): dN/dt = kN(1 − N/M), where M is the carrying capacity. This yields a sigmoidal growth curve that plateaus at N = M.

有限增长（逻辑斯蒂模型）：dN/dt = kN(1 − N/M)，其中 M 为环境容量。其解为一条在 N = M 处趋于平稳的S形增长曲线。

Newton’s law of cooling: dθ/dt = −k(θ − θₑ), where θ is the object temperature and θₑ is the ambient temperature. The solution is θ = θₑ + (θ₀ − θₑ)e^(−kt).

牛顿冷却定律：dθ/dt = −k(θ − θₑ)，其中 θ 为物体温度、θₑ 为环境温度。解为 θ = θₑ + (θ₀ − θₑ)e^(−kt)。

For second-order applications, simple harmonic motion is described by d²x/dt² = −ω²x, whose solution is x = A cos(ωt) + B sin(ωt), or equivalently x = R cos(ωt + φ).

在二阶应用中，简谐运动由 d²x/dt² = −ω²x 描述，其通解为 x = A cos(ωt) + B sin(ωt)，等价形式为 x = R cos(ωt + φ)。

9. Common Pitfalls and Exam Strategies | 常见错误与考试策略

Students frequently lose marks on differential equation questions due to avoidable errors. Being aware of these pitfalls before the exam is half the battle won.

学生常在微分方程题目中因可避免的错误而失分。考前了解这些常见陷阱，等于赢了一半。

Forgetting the constant of integration: always include C after every integration in indefinite integrals.

忘记积分常数：不定积分后务必加上常数 C。
Incorrectly separating variables: if the equation cannot be written as f(y)dy = g(x)dx, do not force it; use the integrating factor instead.

不当分离变量：若方程无法写成 f(y)dy = g(x)dx 的形式，不要强行分离，应改用积分因子法。
Sign errors when substituting the integrating factor: double-check that P(x) is transcribed correctly and the exponent sign matches.

代入积分因子时符号错误：仔细核对 P(x) 是否抄写正确，指数符号是否对应。
Wrong trial function for PI: if the standard trial form is also a solution of the homogeneous equation, multiply by x (or x²) until it is independent of the CF.

特解试探函数选择错误：若标准试探形式恰好是齐次方程的解，则需乘以 x（或 x²）直到与CF线性无关。
Not applying both initial conditions for second-order equations: both y and y′ conditions must be used to determine A and B.

二阶方程未同时利用两个初始条件：必须使用 y 和 y′ 两个条件联立求出 A 和 B。

Exam strategy: always state the auxiliary equation explicitly, show the general solution before applying conditions, and verify the particular solution by substituting back where feasible.

考试策略：始终明确写出辅助方程，先写出通解再应用条件，并在可行时将特解代回验证。

10. Worked Example Combining Multiple Skills | 综合技能例题

Let us now solve a comprehensive problem that integrates several techniques studied above.

下面我们求解一道综合运用以上多种技巧的完整题目。

Problem: Solve the differential equation d²y/dx² − 4y = 8x, subject to y(0) = 2 and y′(0) = 4.

题目：求解微分方程 d²y/dx² − 4y = 8x，满足条件 y(0) = 2 和 y′(0) = 4。

Solution:

解答：

Step 1 — complementary function. The auxiliary equation is λ² − 4 = 0, giving λ = ±2. Hence CF = Ae^(2x) + Be^(−2x).

第一步——互补函数：辅助方程为 λ² − 4 = 0，得 λ = ±2。因此 CF = Ae^(2x) + Be^(−2x)。

Step 2 — particular integral. Since f(x) = 8x is linear, try PI = ax + b. Then d²(PI)/dx² = 0, and substituting gives −4(ax + b) = 8x, so −4a = 8 and −4b = 0, yielding a = −2, b = 0. PI = −2x.

第二步——特解：因 f(x) = 8x 为一次多项式，尝试 PI = ax + b。则 d²(PI)/dx² = 0，代入得 −4(ax + b) = 8x，即 −4a = 8、−4b = 0，解得 a = −2、b = 0。PI = −2x。

Step 3 — general solution: y = Ae^(2x) + Be^(−2x) − 2x.

第三步——通解：y = Ae^(2x) + Be^(−2x) − 2x。

Step 4 — apply conditions. y(0) = A + B = 2. Also y′ = 2Ae^(2x) − 2Be^(−2x) − 2, so y′(0) = 2A − 2B − 2 = 4, giving A − B = 3. Solving the system A + B = 2 and A − B = 3 yields A = 5/2, B = −1/2.

第四步——代入条件：y(0) = A + B = 2。又 y′ = 2Ae^(2x) − 2Be^(−2x) − 2，所以 y′(0) = 2A − 2B − 2 = 4，即 A − B = 3。联立 A + B = 2 与 A − B = 3，得 A = 5/2、B = −1/2。

Final answer:

最终答案：

y = (5/2)e^(2x) − (1/2)e^(−2x) − 2x

This example demonstrates the full workflow: finding the CF, constructing the PI, combining them, and solving for constants using boundary conditions.

该例题展示了完整流程：求互补函数、构造特解、合并为通解，并利用边界条件求解常数。

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Differential Equations Fundamentals and Solution Strategies | 微分方程基础与求解策略