A-Level Mathematics: Methods & Steps for Solving Differential Equations | A-Level 数学：解微分方程的方法与步骤

📚 A-Level Mathematics: Methods & Steps for Solving Differential Equations | A-Level 数学：解微分方程的方法与步骤

Differential equations are a cornerstone of A-Level Mathematics and Further Mathematics. They describe how a quantity changes with respect to another variable, and mastering the standard solution techniques is essential for both pure mathematics questions and real-world modelling problems in exams.

微分方程是 A-Level 数学与进阶数学的核心内容之一。它描述了一个量相对于另一个量的变化规律。无论是纯数学题还是实际建模题，熟练掌握标准的求解方法都是取得高分的关键。

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

A differential equation (DE) is an equation that involves an unknown function and one or more of its derivatives. For example, dy/dx = 3x² + 2 is a first-order DE, while d²y/dx² – 5 dy/dx + 6y = 0 is a second-order DE. The order of a DE is the highest derivative it contains.

微分方程是含有未知函数及其一阶或高阶导数的方程。例如，dy/dx = 3x² + 2 是一阶微分方程，而 d²y/dx² – 5 dy/dx + 6y = 0 是二阶微分方程。方程的阶数等于其中出现的最高阶导数的阶数。

In A-Level Mathematics, you are mainly required to solve first-order differential equations — those containing only dy/dx — using two core techniques: separation of variables and the integrating factor method. Some boards also ask for homogeneous first-order DEs solved by substitution.

在 A-Level 数学中，主要要求求解一阶微分方程（只含 dy/dx 的方程），核心方法有变量分离法与积分因子法。部分考试局还要求掌握用换元法求解齐次一阶微分方程。

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2. Separation of Variables | 变量分离法

If a first-order DE can be written in the form dy/dx = g(x) · f(y), we can separate the variables. The key step is to rearrange so that every y-term is with dy and every x-term is with dx:

若一阶微分方程可以写成 dy/dx = g(x) · f(y) 的形式，我们就可以分离变量。关键步骤是重新整理，使所有含 y 的项与 dy 放在一起，所有含 x 的项与 dx 放在一起：

∫ 1/f(y) dy = ∫ g(x) dx + C

• Rearrange the equation to separate y and x. | 重新整理方程，将 y 与 x 分离到等号两侧。
• Integrate both sides: the left with respect to y, the right with respect to x. | 两边分别积分：左边对 y，右边对 x。
• Add the constant of integration C on one side only. | 只在其中一侧加上积分常数 C。
• Solve for y explicitly if possible; otherwise leave the answer as an implicit equation. | 如可能，尽量解出 y；否则保留为隐式方程。

Worked example: solve dy/dx = 2xy.

典型例题：求解 dy/dx = 2xy。

1/y dy = 2x dx → ∫ 1/y dy = ∫ 2x dx → ln|y| = x² + C

Since C is arbitrary, we may write ln|y| = x² + C and then exponentiate both sides:

由于 C 为任意常数，可将 ln|y| = x² + C 两边取指数：

|y| = e^(x² + C) → y = A e^(x²), where A = ±e^C

The absolute value disappears because the constant A can absorb the positive or negative sign.

绝对值符号可以去掉，因为常数 A 已经包含了正负号。

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3. Integrating Factor Method | 积分因子法

For a first-order linear DE written in the standard form dy/dx + P(x)y = Q(x), we multiply both sides by the integrating factor (IF):

对于标准形式的一阶线性微分方程 dy/dx + P(x)y = Q(x)，在等式两边同时乘以积分因子（IF）：

IF = e^(∫ P(x) dx)

This transforms the left-hand side into the derivative of a product, so the equation becomes:

这样做可以将左边化为一个乘积的导数，方程变为：

d/dx [ y · e^(∫ P(x) dx) ] = Q(x) · e^(∫ P(x) dx)

Then integrate both sides and solve for y. Do not forget the constant of integration.

然后两边积分并解出 y。不要忘记积分常数。

Worked example: solve dy/dx + 2y = e^x, given y(0) = 1.

典型例题：求解 dy/dx + 2y = e^x，已知 y(0) = 1。

Here P(x) = 2, so IF = e^(∫ 2 dx) = e^(2x).

这里 P(x) = 2，所以 IF = e^(∫ 2 dx) = e^(2x)。

d/dx ( y e^(2x) ) = e^x · e^(2x) = e^(3x)

y e^(2x) = ∫ e^(3x) dx = ⅓ e^(3x) + C

y = ⅓ e^x + C e^(-2x)

Using y(0) = 1: 1 = ⅓ + C, so C = ⅔. Therefore the particular solution is y = ⅓ e^x + ⅔ e^(-2x).

代入初值 y(0) = 1：1 = ⅓ + C，得 C = ⅔。因此特解为 y = ⅓ e^x + ⅔ e^(-2x)。

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4. Homogeneous Differential Equations | 齐次微分方程

A first-order DE is homogeneous if it can be expressed as dy/dx = F(y/x). A typical example is dy/dx = (x² + y²)/(xy).

如果一阶微分方程可以写成 dy/dx = F(y/x) 的形式，则称其为齐次微分方程。典型例子如 dy/dx = (x² + y²)/(xy)。

Method: use the substitution y = vx, where v is a new function of x. By the product rule:

解法：令 y = vx，其中 v 是 x 的新函数。由乘法法则可得：

dy/dx = v + x dv/dx

Substituting into dy/dx = F(y/x) gives v + x dv/dx = F(v), which is a separable equation in v and x. Solve it, then replace v with y/x.

代回 dy/dx = F(y/x) 得到 v + x dv/dx = F(v)，这是关于 v 与 x 的可分离变量方程。求出 v 后，再代回 v = y/x。

Worked example: solve dy/dx = (x² + y²)/(xy).

典型例题：求解 dy/dx = (x² + y²)/(xy)。

First rewrite the right-hand side:

首先改写右边：

dy/dx = x/y + y/x = 1/v + v

With y = vx, we get:

令 y = vx，得到：

v + x dv/dx = v + 1/v → x dv/dx = 1/v → v dv = dx/x

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