📚 Introduction to Differential Equations | 微分方程导论
A differential equation is an equation that connects an unknown function with one or more of its derivatives. These equations appear throughout A-level Mathematics because they describe how quantities change, from population growth to the temperature of a cooling object.
微分方程是将未知函数与其一个或多个导数联系起来的方程。这类方程贯穿 A-level 数学，因其能描述各种量的变化，例如人口增长或物体冷却时的温度变化。
1. What Is a Differential Equation? | 什么是微分方程
A differential equation involves an unknown function, usually written y(x), and at least one derivative such as dy/dx or d²y/dx². For example, the equation dy/dx = 3x² + 2 is a differential equation because it contains the first derivative dy/dx.
微分方程包含一个未知函数，通常记为 y(x)，以及至少一个导数，例如 dy/dx 或 d²y/dx²。例如，方程 dy/dx = 3x² + 2 就是一个微分方程，因为它含有导数 dy/dx。
dy/dx = 3x² + 2
This equation tells us that the gradient of the curve y(x) is given by 3x² + 2 at every point. Solving it means finding all functions y whose derivative equals that expression.
该方程告诉我们，曲线 y(x) 在每一点的斜率都由 3x² + 2 给出。求解它意味着找出所有导数等于该表达式的函数 y。
Another common form is d²y/dx² + 5 dy/dx + 6y = 0. This contains a second derivative, so it is a second-order differential equation.
另一种常见形式是 d²y/dx² + 5 dy/dx + 6y = 0。它含有二阶导数，因此是一个二阶微分方程。
2. Order and Degree | 阶与次
The order of a differential equation is the order of the highest derivative that appears in it. The degree is the power of the highest derivative after the equation has been written as a polynomial in derivatives, without fractions or radicals involving derivatives.
微分方程的阶是其中出现的最高阶导数的阶数。次是将方程写成关于导数的多项式、且不含导数分式或根式后，最高阶导数的次数。
(dy/dx)² + 3y = x
This equation is first order because the highest derivative is dy/dx. Its degree is 2 because the highest derivative is raised to the power 2.
这个方程是一阶的，因为最高导数是 dy/dx。它的次为 2，因为最高导数被提升到 2 次幂。
At A-level, most exam questions focus on first-order equations. Second-order equations are usually only met in the context of verifying a solution or in further mathematics.
在 A-level 考试中，大多数题目集中在一阶方程。二阶方程通常只在验证解的背景中出现，或属于进阶数学内容。
3. General and Particular Solutions | 通解与特解
When we solve a differential equation, we usually obtain a family of functions containing at least one arbitrary constant. This is called the general solution. For a first-order equation, there is one arbitrary constant.
解微分方程时，通常得到一个包含至少一个任意常数的函数族，称为通解。对于一阶方程，通解中有一个任意常数。
dy/dx = 3x² + 2 → y = x³ + 2x + C
Here C is the arbitrary constant. If we also know that the curve passes through a particular point, such as y(0) = 5, we can find C. Substituting x = 0 gives C = 5, so the particular solution is y = x³ + 2x + 5.
这里 C 是任意常数。如果我们还知道曲线经过某个特定点，例如 y(0) = 5，就可以求出 C。代入 x = 0 得到 C = 5，因此特解为 y = x³ + 2x + 5。
The extra condition y(0) = 5 is called an initial condition or boundary condition. In AQA questions, you will often be given such a condition to determine the constant.
附加条件 y(0) = 5 称为初始条件或边界条件。AQA 试题中通常会给出这样一个条件来确定常数。
4. Verifying a Solution | 验证解
To show that a given function is a solution of a differential equation, differentiate the function as many times as needed and substitute the result back into the equation. The left-hand side and right-hand side must be identical.
要证明一个给定函数是微分方程的解，需要对函数进行足够次数的求导，并将结果代回方程。左边和右边必须完全相同。
Example: show that y = A exp(2x) satisfies dy/dx = 2y.
例子：证明 y = A exp(2x) 满足 dy/dx = 2y。
y = A exp(2x) → dy/dx = 2A exp(2x) = 2y
Since the derivative is exactly 2y, the function satisfies the differential equation for every constant A. This is a useful check in exam solutions.
因为导数恰好等于 2y，所以该函数对任意常数 A 都满足微分方程。这是考试中常用的检验方法。
5. Separable First-Order Equations | 可分离变量的一阶方程
Many first-order differential equations in AQA Mathematics are separable. This means they can be written in the form dy/dx = f(x)g(y), where f(x) depends only on x and g(y) depends only on y.
AQA 数学中的许多一阶微分方程都是可分离变量的。这意味着它们可以写成 dy/dx = f(x)g(y) 的形式，其中 f(x) 只依赖于 x，g(y) 只依赖于 y。
dy/dx = 2xy
Here f(x) = 2x and g(y) = y. We can separate the variables by dividing both sides by y and multiplying both sides by dx, giving 1/y dy = 2x dx.
这里 f(x) = 2x，g(y) = y。我们可以通过两边除以 y 并两边乘以 dx 来分离变量，得到 1/y dy = 2x dx。
We must be careful: separation by division assumes y ≠ 0. After solving, we check whether y = 0 is also a solution of the original equation.
需要注意：通过除法分离变量时假设 y ≠ 0。求解后还要检查 y = 0 是否也是原方程的解。
6. The Method of Separation of Variables | 分离变量法步骤
The method of separation of variables follows a clear sequence. AQA examiners expect each line to be shown clearly.
分离变量法遵循清晰的步骤。AQA 阅卷人希望每一步都清楚写出来。
- Step 1: Write the equation in the form dy/dx = f(x)g(y). | 第一步：将方程写成 dy/dx = f(x)g(y) 的形式。
- Step 2: Separate the variables to get 1/g(y) dy = f(x) dx. | 第二步：分离变量，得到 1/g(y) dy = f(x) dx。
- Step 3: Integrate both sides. | 第三步：两边同时积分。
- Step 4: Include one constant of integration on one side only. | 第四步：只在一边加一个积分常数。
- Step 5: Solve for y explicitly if possible, then apply any given condition. | 第五步：如果可能，解出 y 的显式表达式，再代入给定条件。
For example, solving dy/dx = 2xy gives ∫ 1/y dy = ∫ 2x dx, so ln|y| = x² + C. From here we can exponentiate to solve for y.
例如，求解 dy/dx = 2xy 得到 ∫ 1/y dy = ∫ 2x dx，因此 ln|y| = x² + C。由此可以通过指数运算解出 y。
7. Finding Particular Solutions with Initial Conditions | 用初始条件求特解
Worked example: solve the differential equation dy/dx = 2xy given that y(0) = 3.
例题：求解微分方程 dy/dx = 2xy，已知 y(0) = 3。
First separate the variables.
首先分离变量。
1/y dy = 2x dx
Now integrate both sides.
现在两边积分。
ln|y| = x² + C
Exponentiate to remove the logarithm.
通过指数运算消去对数。
|y| = exp(x² + C) = exp(C) exp(x²)
Let A = exp(C), so y = A exp(x²). Now use the initial condition y(0) = 3.
令 A = exp(C)，则 y = A exp(x²)。现在使用初始条件 y(0) = 3。
3 = A exp(0) = A → A = 3
The particular solution is y = 3 exp(x²). Always substitute your final answer back into the original equation to verify it.
特解为 y = 3 exp(x²)。最后一定要把答案代回原方程进行验证。
8. Modelling with Differential Equations | 用微分方程建模
AQA questions often describe a rate of change in words. You need to translate the statement into a differential equation. For example, “the rate of change of P is proportional to P” becomes dP/dt = kP.
AQA 题目经常用文字描述变化率。你需要把这些语句翻译成微分方程。例如，”P 的变化率与 P 成正比” 可以写成 dP/dt = kP。
dP/dt = kP → P = P₀ exp(kt)
Here P₀ is the initial population or quantity at time t = 0, and k is the growth constant. This model describes exponential growth when k > 0 and exponential decay when k < 0.
这里 P₀ 是时间 t = 0 时的初始数量，k 是增长常数。该模型在 k > 0 时描述指数增长，在 k < 0 时描述指数衰减。
| Exponential growth | 指数增长 | dP/dt = kP |
| Newton cooling | 牛顿冷却 | dT/dt = -k(T –
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