Differential Equations | 微分方程

📚 Differential Equations | 微分方程

Differential equations are central to A-Level mathematics. They express how a quantity changes with respect to another variable, often time.

微分方程是 A-Level 数学的核心内容。它们描述一个量相对于另一个变量（通常是时间）的变化方式。

---

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

A differential equation is an equation that contains derivatives. The order of a differential equation is the highest derivative that appears in it.

微分方程是含有导数的方程。方程的阶数是方程中出现的最高阶导数的阶数。

For example, dy/dx = 3x is a first-order differential equation, while d²y/dx² + y = 0 is a second-order differential equation.

例如，dy/dx = 3x 是一阶微分方程，而 d²y/dx² + y = 0 是二阶微分方程。

dy/dx = 3x → first order

d²y/dx² + y = 0 → second order

---

2. Separation of Variables | 变量分离法

If a differential equation can be written in the form dy/dx = f(x)g(y), we can separate the variables by moving all y-terms to the left and all x-terms to the right.

如果一个微分方程可以写成 dy/dx = f(x)g(y) 的形式，我们可以通过变量分离法，把所有含 y 的项移到左边，把所有含 x 的项移到右边。

∫ 1/g(y) dy = ∫ f(x) dx

Once both sides are integrated, we solve for y if possible. This is why the method is called separation of variables.

两边积分后，若可能则解出 y。这就是“变量分离法”名称的由来。

For example, solve dy/dx = 2xy.

例如，求解 dy/dx = 2xy。

Separating gives:

分离变量得到：

(1/y) dy = 2x dx

Integrating both sides:

两边积分：

ln|y| = x² + C

Therefore the general solution is y = A e^(x²), where A = ±e^C.

因此通解为 y = A e^(x²)，其中 A = ±e^C。

---

3. Integrating Factor | 积分因子法

For a linear first-order differential equation of the form dy/dx + P(x)y = Q(x), we cannot always separate the variables. Instead, we use an integrating factor.

对于形如 dy/dx + P(x)y = Q(x) 的一阶线性微分方程，我们不一定能分离变量。此时可以使用积分因子法。

Integrating factor: I = e^(∫ P(x) dx)

Multiply every term by I. The left side becomes the derivative of Iy:

将每一项都乘以 I。左边就会变成 Iy 的导数：

d/dx (Iy) = I Q(x)

Then integrate both sides with respect to x.

然后两边对 x 积分。

Example: solve dy/dx + (2/x)y = x.

例：求解 dy/dx + (2/x)y = x。

Here P(x) = 2/x, so the integrating factor is:

这里 P(x) = 2/x，因此积分因子为：

I = e^(∫ 2/x dx) = e^(2 ln|x|) = x²

Multiplying through by x² gives:

两边乘以 x² 得：

d/dx (x²y) = x³

Integrate:

积分：

x²y = x⁴/4 + C

So the general solution is:

因此通解为：

y = x²/4 + C x⁻²

---

4. Boundary and Initial Conditions | 边界条件与初始条件

The general solution contains an arbitrary constant C. A boundary condition or initial condition gives extra information that allows us to find C and obtain a particular solution.

通解中含有一个任意常数 C。边界条件或初始条件提供了额外信息，使我们能求出 C，从而得到特解。

Using the previous example, suppose y(1) = 2. Substitute x = 1 and y = 2 into y = x²/4 + C x⁻².

继续使用上面的例子，假设 y(1) = 2。将 x = 1 和 y = 2 代入 y = x²/4 + C x⁻²。

2 = 1/4 + C

C = 7/4

Therefore the particular solution is:

因此特解为：

y = x²/4 + 7/(4x²)

Always check that the given condition is substituted after integration, not before.

务必在积分之后代入给定条件，而不是在积分之前代入。

---

5. Exponential Growth and Decay | 指数增长与衰减

The simplest model for growth and decay is the equation dP/dt = kP. This says that the rate of change of P is proportional to P itself.

增长与衰减最简单的模型是方程 dP/dt = kP。它表示 P 的变化率与 P 自身成正比。

The general solution is:

其通解为：

P = P₀ e^(kt)

where P₀ is the initial value at t = 0.

其中 P₀ 是 t = 0 时的初始值。

If k > 0, the quantity grows exponentially. If k < 0, it decays exponentially.

如果 k > 0，则数量指数增长；如果 k < 0，则数量指数衰减。

For radioactive decay, the half-life T satisfies P = P₀/2, giving:

对于放射性衰变，半衰期 T 满足 P = P₀/2，因此：

T = ln 2 / (−k)

This formula is useful in both pure mathematics and applied problems.

这个公式在纯数学和应用题中都非常有用。

---

6. Modelling with Differential Equations | 微分方程建模

Real-life situations can often be translated directly into differential equations using rate language.

现实生活中的许多情境可以直接用“变化率”的语言翻译成微分方程。

For example, Newton’s law of cooling states that the rate of change of temperature is proportional to the difference between the object’s temperature and the surrounding temperature.

例如，牛顿冷却定律指出：物体温度的变化率与物体温度和周围环境温度之差成正比。

dT/dt = −k(T − Tₛ)

Here T is the object temperature, Tₛ is the surrounding temperature, and k is a positive constant. The negative sign indicates cooling.

其中 T 是物体温度，Tₛ 是环境温度，k 为正数。负号表示物体在降温。

Because T − Tₛ appears in the equation, the variables can be separated:

因为方程中出现 T − Tₛ，所以可以分离变量：

∫ 1/(T − Tₛ) dT = ∫ −k dt

ln|T − Tₛ| = −kt + C

This gives T = Tₛ + A e^(−kt).

因此 T = Tₛ + A e^(−kt)。

---

7. Forming the Equation from a Rate Statement | 从速率表述建立方程

Exam questions often give a verbal statement and ask you to form a differential equation. Look for phrases such as “rate of change”, “proportional to”, and “inversely proportional to”.

考试题经常给出一段文字描述，要求你建立微分方程。请注意“变化率”“成正比”“成反比”等关键词。

Rate statement	Differential equation
The population increases at a rate proportional to the population.	dP/dt = kP
The temperature decreases at a rate proportional to the temperature difference.	dT/dt = −k(T − Tₐ)
The volume of water leaks out at a rate proportional to the square root of the volume.	dV/dt = −k√V

It is important to identify whether the quantity is increasing or decreasing. If it is decreasing, include a negative sign.

注意判断该量是在增加还是减少。若在减少，需要加上负号。

---

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）