Solving Differential Equations | 求解微分方程

📚 Solving Differential Equations | 求解微分方程

Differential equations lie at the heart of calculus and appear across mathematics, physics, engineering and economics. In A-Level Maths (Edexcel), you are expected to solve first-order separable differential equations, apply initial conditions to find particular solutions, and construct simple differential equations from real-world contexts.

微分方程是微积分的核心，广泛应用于数学、物理、工程和经济学。在A-Level数学（Edexcel）中，你需要掌握一阶可分离变量微分方程的求解、利用初始条件确定特解，以及根据实际问题建立简单的微分方程。

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

A differential equation is any equation that contains one or more derivatives. The order of a differential equation is determined by the highest derivative present. For instance, dy/dx = 5x is a first-order differential equation because the highest derivative is the first derivative.

微分方程是包含一个或多个导数的方程。方程的“阶”由出现的最高阶导数决定。例如 dy/dx = 5x 是一个一阶微分方程，因为最高阶导数为一阶导数。

In Edexcel A-Level, you will only deal with first-order ordinary differential equations (ODEs) that can be solved by separating the variables.

在Edexcel A-Level考试中，你只会遇到可以通过分离变量法求解的一阶常微分方程（ODE）。

2. The Form dy/dx = f(x)g(y) | 形如 dy/dx = f(x)g(y) 的方程

Separable differential equations are those that can be written in the form dy/dx = f(x)g(y), where the right-hand side is a product of a function of x only and a function of y only.

可分离变量的微分方程是指能够写成 dy/dx = f(x)g(y) 形式的一阶微分方程，其右边只是 x 的函数与只是 y 的函数的乘积。

For example, dy/dx = 3x²y can be seen as f(x) = 3x² and g(y) = y. Recognising this structure is the first step towards solving the equation.

例如，dy/dx = 3x²y 可以看作 f(x) = 3x²，g(y) = y。识别出这个结构是解方程的第一步。

3. The Separation of Variables Technique | 分离变量法

The method involves rearranging the equation so that all terms involving y appear on one side with dy, and all terms involving x appear on the other side with dx, before integrating both sides.

分离变量法通过重新整理方程，让所有包含 y 的项与 dy 在一边，所有包含 x 的项与 dx 在另一边，然后两边同时积分。

If dy/dx = f(x)g(y), we rewrite as (1/g(y)) dy = f(x) dx, provided g(y) ≠ 0. Then integrate to obtain the general solution: ∫ (1/g(y)) dy = ∫ f(x) dx + C.

如果 dy/dx = f(x)g(y)，在 g(y) ≠ 0 的条件下，可重写为 (1/g(y)) dy = f(x) dx。然后积分得到通解：∫ (1/g(y)) dy = ∫ f(x) dx + C。

4. Worked Example 1: Basic Separation | 例题1：基本分离变量

Solve dy/dx = 2xy. Step 1: Separate variables. Divide both sides by y and multiply by dx: (1/y) dy = 2x dx.

求解 dy/dx = 2xy。第一步：分离变量。两边除以 y 并乘以 dx：(1/y) dy = 2x dx。

Step 2: Integrate both sides. ∫ (1/y) dy = ∫ 2x dx gives ln|y| = x² + C.

第二步：两边积分。∫ (1/y) dy = ∫ 2x dx 得到 ln|y| = x² + C。

Step 3: Solve for y. Exponentiate both sides: |y| = e^(x²+C) = e^C · e^(x²). Let A = ±e^C, then y = A e^(x²), where A is an arbitrary constant.

第三步：解出 y。两边取指数：|y| = e^(x²+C) = e^C · e^(x²)。令 A = ±e^C，则通解为 y = A e^(x²)，A 为任意常数。

5. Using Initial or Boundary Conditions | 利用初始条件或边界条件

If you are given an additional condition, such as y = 3 when x = 0, you can substitute these values into the general solution to find the particular constant A, thus obtaining a particular solution.

如果题目给出了附加条件，比如当 x = 0 时 y = 3，你可以把这些值代入通解，求出具体的常数 A，从而得到特解。

Using the above example: 3 = A e⁰ ⇒ A = 3. Hence the particular solution is y = 3 e^(x²). Edexcel exam questions almost always ask for the particular solution.

沿用上例：3 = A e⁰ ⇒ A = 3。因此特解为 y = 3 e^(x²)。Edexcel 考题几乎总是要求求出特解。

Always check that the condition is applied correctly, and remember that the constant of integration should be determined only after the integration has been performed.

务必确认条件使用正确，并记住积分常数应在积分完成后才能确定。

6. Worked Example 2: Trigonometric and Logarithmic Integrals | 例题2：含三角函数与对数的积分

Find the particular solution of dy/dx = y cos x, given that y = 2 when x = π/2.

求 dy/dx = y cos x 的特解，已知当 x = π/2 时 y = 2。

Separate: (1/y) dy = cos x dx. Integrate: ∫ (1/y) dy = ∫ cos x dx ⇒ ln|y| = sin x + C.

分离变量：(1/y) dy = cos x dx。积分：∫ (1/y) dy = ∫ cos x dx ⇒ ln|y| = sin x + C。

Substitute y = 2, x = π/2: ln 2 = sin(π/2) + C = 1 + C ⇒ C = ln 2 − 1.

代入 y = 2, x = π/2：ln 2 = sin(π/2) + C = 1 + C ⇒ C = ln 2 − 1。

Thus ln|y| = sin x + ln 2 − 1. Exponentiate: |y| = e^(sin x + ln 2 − 1) = 2 e^(sin x − 1). So y = 2 e^(sin x − 1) (retaining the positive solution as given by the initial condition).

因此 ln|y| = sin x + ln 2 − 1。取指数：|y| = e^(sin x + ln 2 − 1) = 2 e^(sin x − 1)。因为初值给出的是正值，特解为 y = 2 e^(sin x − 1)。

7. Integrating Factors – Not Required for Edexcel A-Level | 积分因子法 – Edexcel A-Level 不做要求

At this level, you only need the method of separation of variables. The integrating factor method is part of Further Mathematics. It is essential not to confuse the two.

在当前阶段，你只需要掌握分离变量法。积分因子法是进阶数学的内容。切勿将两者混淆。

Edexcel exam questions will always be designed so that the differential equation is either already separable or can easily be turned into a separable form.

Edexcel 考试题目的设计总能保证微分方程已经是可分离的，或可以通过简单变换化为可分离形式。

8. Common Domain Restrictions and Singular Solutions | 常见定义域限制与奇解

When dividing by g(y), you must state g(y) ≠ 0 explicitly. Edexcel often expects you to mention that the particular solution satisfies this condition throughout the given domain.

在除以 g(y) 时，必须明确 g(y) ≠ 0。Edexcel 通常要求你指出特解在整个给定定义域内都满足该条件。

For example, if dy/dx = 2xy and you divide by y, you lose the trivial solution y = 0. It is good practice to note that y = 0 is also a solution (a singular solution) but is usually excluded when an initial condition y ≠ 0 is provided.

例如，对于 dy/dx = 2xy，若除以 y，你会丢失平凡解 y = 0。良好的习惯是指出 y = 0 也是一个解（奇解），但题目给出初值 y ≠ 0 时通常不予考虑。

9. Forming Differential Equations from Worded Problems | 根据文字问题建立微分方程

Modelling questions ask you to translate a description of a rate of change into a differential equation. Typical contexts include population growth, radioactive decay, cooling, and mixing problems.

建模题要求你将变化率的文字描述转化为微分方程。常见情境包括人口增长、放射性衰变、冷却和混合问题。

For instance, “the rate of increase of a population P is proportional to P” leads to dP/dt = kP. “The temperature T of an object decreases at a rate proportional to the excess over room temperature R” yields dT/dt = −k(T − R).

例如，“人口 P 的增长率与 P 成正比”可导出 dP/dt = kP。“物体温度 T 的冷却速率与超过室温 R 的部分成正比”可得到 dT/dt = −k(T − R)。

Once you have formed the differential equation, you can solve it using separation of variables and apply any given data to determine the constants.

一旦建立了微分方程，就可以使用分离变量法求解，并利用给出的数据确定常数。

10. Worked Example 3: Exponential Growth with Data | 例题3：带数据的指数增长

The number of bacteria N in a culture grows at a rate proportional to N. Initially N = 500, and after 2 hours N = 2000. Find N as a function of time t in hours.

培养皿中细菌数量 N 的增长速率与 N 成正比。初始时刻 N = 500，2 小时后 N = 2000。求 N 关于时间 t（小时）的函数。

The differential equation is dN/dt = kN. Separate: (1/N) dN = k dt. Integrate: ln|N| = k t + C ⇒ N = A e^(kt), where A = e^C.

微分方程为 dN/dt = kN。分离变量：(1/N) dN = k dt。积分：ln|N| = k t + C ⇒ N = A e^(kt)，其中 A = e^C。

At t = 0, N = 500 ⇒ 500 = A e⁰ ⇒ A = 500. At t = 2, N = 2000 ⇒ 2000 = 500 e^(2k) ⇒ e^(2k) = 4 ⇒ 2k = ln 4 ⇒ k = ½ ln 4 = ln 2.

当 t = 0, N = 500 ⇒ 500 = A e⁰ ⇒ A = 500。当 t = 2, N = 2000 ⇒ 2000 = 500 e^(2k) ⇒ e^(2k) = 4 ⇒ 2k = ln 4 ⇒ k = ½ ln 4 = ln 2。

Hence N = 500 e^(t ln 2) = 500 · (e^(ln 2))^t = 500 · 2^t. The number doubles every hour.

因此 N = 500 e^(t ln 2) = 500 · (e^(ln 2))^t = 500 · 2^t。细菌数量每小时翻一番。

11. Important Integral Forms to Recognise | 需要熟悉的重要积分形式

When solving differential equations, you will frequently encounter integrals like ∫ 1/(ax+b) dx, ∫ e^(kx) dx, ∫ sin x dx, ∫ cos x dx, and ∫ sec² x dx. These must be known confidently from the Pure specification.

求解微分方程时，你会频繁遇到诸如 ∫ 1/(ax+b) dx、∫ e^(kx) dx、∫ sin x dx、∫ cos x dx 和 ∫ sec² x dx 等积分。务必熟练掌握纯数部分要求的这些积分。

A quick reference table may help:

Integral	Result
∫ (1/x) dx	ln\|x\| + C
∫ e^(kx) dx	(1/k) e^(kx) + C
∫ cos x dx	sin x + C
∫ sin x dx	−cos x + C
∫ sec² x dx	tan x + C
∫ 1/(ax+b) dx	(1/a) ln\|ax+b\| + C

熟练掌握这些基本积分能让分离变量后的积分过程更顺畅。

12. Common Mistakes and Tips for Edexcel Exams | 常见错误与 Edexcel 考试技巧

Mistake 1: Forgetting the constant of integration. Always include +C immediately after integrating. Many marks are lost by omitting it and then struggling to find the particular solution correctly.

错误1：忘记加积分常数。每次积分后务必立刻写上 +C。漏掉常数会导致无法正确求出特解，丢分严重。

Mistake 2: Not rewriting logarithmic constants appropriately. For instance, ln|y| = 2 ln|x| + C can be simplified to ln|y| = ln(x²) + C, then y = A x² with A = ±e^C. Avoid leaving A as e^C inside a logarithm.

错误2：没有适当整理对数常数。例如 ln|y| = 2 ln|x| + C 可化简为 ln|y| = ln(x²) + C，然后 y = A x²，其中 A = ±e^C。不要将 A 留在对数里。

Mistake 3: Mixing up signs when integrating trigonometric functions. ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C. Double-check signs.

错误3：三角函数积分时搞错正负号。∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C。请仔细核对正负号。

Edexcel often examines your ability to express the final answer in a tidy form – factor out constants and simplify expressions using index laws.

Edexcel 经常考察你将最终答案整理成简洁形式的能力——提出常数、利用指数法则化简表达式都是得分点。

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Solving Differential Equations | 求解微分方程