Introduction to Differential Equations | 微分方程导论

📚 Introduction to Differential Equations | 微分方程导论

A differential equation is a mathematical equation that relates a function to its derivatives. In A-Level mathematics, we use differential equations to model situations involving rates of change, such as population growth, cooling, and motion.

微分方程是联系函数及其导数的数学方程。在 A-Level 数学中，我们用微分方程来建模涉及变化率的情境，例如人口增长、冷却和运动。

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

An equation that contains at least one derivative is called a differential equation. For example, dy/dx = 2x is a differential equation.

包含至少一个导数的方程称为微分方程。例如，dy/dx = 2x 就是一个微分方程。

The derivative dy/dx represents the instantaneous rate of change of y with respect to x. Solving a differential equation means finding a function y(x) that satisfies the relationship.

导数 dy/dx 表示 y 关于 x 的瞬时变化率。求解微分方程就是找出满足该关系的函数 y(x)。

2. Order and Degree | 阶与次数

The order of a differential equation is the highest derivative present. The degree is the power of the highest derivative after the equation has been made rational in the derivatives.

微分方程的阶是方程中出现的最高阶导数。次数是在方程对导数化为有理式后最高阶导数的幂次。

For example, d²y/dx² + 3(dy/dx)² + y = 0 is a second-order, first-degree equation because the highest derivative is d²y/dx² and its power is 1.

例如，d²y/dx² + 3(dy/dx)² + y = 0 是一个二阶、一次方程，因为最高阶导数是 d²y/dx²，且其幂次为 1。

First-order equation: dy/dx + y = x

一阶方程：dy/dx + y = x
Second-order equation: d²y/dx² + y = 0

二阶方程：d²y/dx² + y = 0

Order = highest derivative; Degree = power of that derivative.

阶 = 最高阶导数；次数 = 该导数的幂次。

3. General and Particular Solutions | 通解与特解

The general solution of a first-order differential equation contains one arbitrary constant C. This reflects the fact that integrating introduces a constant.

一阶微分方程的通解包含一个任意常数 C。这反映了积分会引入一个常数。

A particular solution is obtained when an initial condition or boundary condition is used to determine the constant C.

当使用初始条件或边界条件确定常数 C 时，得到特解。

For example, the general solution of dy/dx = 2x is y = x² + C. If y(1) = 3, then C = 2, so the particular solution is y = x² + 2.

例如，dy/dx = 2x 的通解是 y = x² + C。若 y(1) = 3，则 C = 2，所以特解为 y = x² + 2。

4. Solving First-Order Differential Equations by Separation of Variables | 用分离变量法求解一阶微分方程

If a differential equation can be written in the form dy/dx = f(x)g(y), we can separate the variables:

如果微分方程可以写成 dy/dx = f(x)g(y) 的形式，我们可以分离变量：

∫ 1/g(y) dy = ∫ f(x) dx

This method works because we can treat dy and dx as differentials and rearrange.

这种方法可行，因为我们可以将 dy 和 dx 看作微分项进行重排。

Example: Solve dy/dx = x/y. Rearrange to y dy = x dx. Integrate both sides:

例：求 dy/dx = x/y 的通解。移项得到 y dy = x dx。两边积分：

∫ y dy = ∫ x dx ⇒ y²/2 = x²/2 + C ⇒ x² – y² = constant

Always include the constant of integration.

始终要加上积分常数。

5. Particular Solutions with Initial Conditions | 带初始条件的特解

Suppose we solve a differential equation and obtain a general solution. To find the particular solution, we substitute the given values of x and y to calculate C.

假设我们求解微分方程得到通解。要得到特解，我们将给定的 x 和 y 值代入，计算出 C。

Worked example: Solve dy/dx = 2xy, given y = 1 when x = 0.

例：求微分方程 dy/dx = 2xy 的解，已知 x = 0 时 y = 1。

Separate variables: (1/y) dy = 2x dx.

分离变量：(1/y) dy = 2x dx。

∫ (1/y) dy = ∫ 2x dx ⇒ ln|y| = x² + C

Using y = 1 when x = 0, we get ln 1 = 0 + C, so C = 0. Thus ln|y| = x², so y = e^(x²).

利用 x = 0 时 y = 1，得到 ln 1 = 0 + C，所以 C = 0。因此 ln|y| = x²，所以 y = e^(x²)。

6. Forming Differential Equations from Real-World Situations | 从实际情境建立微分方程

Many physical situations are modelled by stating that a rate of change is proportional to some quantity.

许多物理情境可以通过“变化率正比于某个量”来建模。

Example 1: Population growth. The rate of change of population P with time t is proportional to P:

例 1：人口增长。人口 P 关于时间 t 的变化率与 P 成正比：

dP/dt = kP

Example 2: Newton’s law of cooling. The rate of change of temperature T is proportional to the difference between T and room temperature Tₛ:

例 2：牛顿冷却定律。温度 T 的变化率与 T 和室温 Tₛ 之差成正比：

dT/dt = -k(T – Tₛ)

The negative sign indicates cooling towards the surrounding temperature.

负号表示温度向周围环境温度降低。

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

Solving dP/dt = kP by separating variables gives the general solution P = A e^(kt). If k > 0, this is exponential growth; if k < 0, it is exponential decay.

通过分离变量求解 dP/dt = kP，得到通解 P = A e^(kt)。若 k > 0，则为指数增长；若 k < 0，则为指数衰减。

Proof:

推导：

dP/dt = kP ⇒ ∫ (1/P) dP = ∫ k dt ⇒ ln|P| = kt + C ⇒ P = A e^(kt)

Here A = e^(C) is the initial value P(0).

其中 A = e^(C) 是初始值 P(0)。

For radioactive decay, the half-life t½ satisfies 1/2 = e^(-k t½), so t½ = ln2 / k.

对于放射性衰变，半衰期 t½ 满足 1/2 = e^(-k t½)，因此 t½ = ln2 / k。

8. Second-Order Differential Equations (Brief Introduction) | 二阶微分方程简介

In A-Level further mathematics or certain questions, you may encounter linear second-order differential equations with constant coefficients:

在 A-Level 进阶数学或某些问题中，你可能会遇到常系数线性二阶微分方程：

a d²y/dx² + b dy/dx + cy = 0

The general solution depends on the roots of the auxiliary equation a m² + b m + c = 0.

通解取决于辅助方程 a m² + b m + c = 0 的根。

If the roots m₁ and m₂ are real and distinct, then y = A e^(m₁x) + B e^(m₂x). If there is a repeated root m, then y = (A + Bx) e^(mx). If the roots are complex, the solution involves sine and cosine.

若根 m₁ 和 m₂ 为相异实根，则 y = A e^(m₁x) + B e^(m₂x)。若有重根 m，则 y = (A + Bx) e^(mx)。若根为复数，则解涉及正弦和余弦。

9. Verification of Solutions | 验证解

Always check a proposed solution by differentiating and substituting back into the original differential equation.

始终通过求导并代回原微分方程来验证所提出的解。

Example: Verify that y = sin(2x) is a solution of d²y/dx² + 4y = 0.

例：验证 y = sin(2x) 是 d²y/dx² + 4y = 0 的解。

dy/dx = 2cos(2x), d²y/dx² = -4sin(2x). Substituting: -4sin(2x) + 4sin(2x) = 0. Verified.

dy/dx = 2cos(2x)，d²y/dx² = -4sin(2x)。代入：-4sin(2x) + 4sin(2x) = 0。验证成立。

10. Common Exam Tips | 常见考试技巧

Always state the constant of integration C.

始终写出积分常数 C。
When separating variables, keep all y terms on one side and all x terms on the other.

分离变量时，将含 y 的项放在一边，含 x 的项放在另一边。
If the initial condition is given, use it immediately after integration to find C.

如果给出初始条件，在积分后立即使用它求出 C。
Check whether your solution satisfies the differential equation.

检查你的解是否满足原微分方程。
Be careful with absolute values and domain restrictions when integrating 1/y.

对 1/y 积分时，注意绝对值符号和定义域限制。
Read if the question asks for a particular solution or general solution.

仔细阅读题目要求的是特解还是通解。

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Introduction to Differential Equations | 微分方程导论