Modelling with Differential Equations | 微分方程建模

📚 Modelling with Differential Equations | 微分方程建模

Differential equations provide a powerful language for describing how quantities change. In A-Level Edexcel Mathematics, modelling with differential equations often means translating a real-world statement about rates into an equation such as dy/dx = k y, solving it by separation of variables, and interpreting the arbitrary constant using initial conditions.

微分方程提供了描述量如何变化的强大语言。在 A-Level Edexcel 数学中，微分方程建模通常意味着将关于变化率的现实语句转化为如 dy/dx = k y 的方程，通过分离变量法求解，并利用初始条件解释任意常数。

1. From Words to Differential Equations | 从文字到微分方程

Many modelling questions start with the phrase ‘the rate of change of … is proportional to …’. This translates directly to dy/dx = k × (expression), where k is the constant of proportionality.

许多建模题以“某量的变化率与……成正比”开头。这直接翻译为 dy/dx = k × (表达式)，其中 k 是比例常数。

When reading a model, highlight phrases like ‘rate of change’, ‘proportional to’, ‘inversely proportional to’, and ‘difference between’. These phrases tell you what belongs on the right-hand side.

阅读模型时，圈出“变化率”“与……成正比”“与……成反比”“两者之差”等短语。这些短语告诉你右侧应写什么。

For example, ‘the rate of cooling is proportional to the excess temperature’ becomes dθ/dt = −k(θ − θₛ), where θₛ is the surrounding temperature.

例如，“冷却速率与超出温度成正比”变为 dθ/dt = −k(θ − θₛ)，其中 θₛ 是环境温度。

2. Direct Proportion and Rates of Change | 正比与变化率

Direct proportion is expressed using a constant k. The units of k depend on the equation: in dP/dt = kP, k has units of time⁻¹, while in dy/dx = kx², k has units of y per x³.

正比关系用常数 k 表示。k 的单位取决于方程：在 dP/dt = kP 中 k 的单位是时间⁻¹，而在 dy/dx = kx² 中 k 的单位是 y/x³。

Always decide whether k should be positive or negative from the direction of change. If the quantity increases when the other increases, k > 0; if it decreases, k < 0 or write a minus sign explicitly.

始终根据变化方向判断 k 应为正还是负。若一量随另一量增大而增大，则 k > 0；若减小，则 k < 0 或明确写出负号。

In many A-Level problems k will be found from data rather than given directly, so keep the equation in terms of k until you can use initial conditions.

在许多 A-Level 问题中，k 需由数据求出而非直接给出，因此先保留含 k 的方程，直到可用初始条件求出。

3. Separation of Variables: The Core Technique | 分离变量法：核心技巧

If dy/dx = f(x)g(y), we can separate the variables: ∫ 1/g(y) dy = ∫ f(x) dx. Do not forget the constant of integration, which can often be written as a single constant on one side.

若 dy/dx = f(x)g(y)，我们可以分离变量：∫ 1/g(y) dy = ∫ f(x) dx。不要忘记积分常数，通常可将其写在一侧作为单个常数。

For dy/dx = ky, separation gives ln|y| = kx + c, so y = A eᵏˣ, where A = ±e

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