Differential Equations of the Form dy/dx = f(x) | 形如 dy/dx = f(x) 的微分方程

📚 Differential Equations of the Form dy/dx = f(x) | 形如 dy/dx = f(x) 的微分方程

Differential equations are central to A-Level Mathematics and real-world modelling. They describe how a quantity changes with respect to another. In this article, we focus on first-order differential equations of the form dy/dx = f(x), and then extend the same ideas to related forms such as dy/dx = g(y) and separable variables. We will cover the general solution, particular solutions, applications, and numerical methods.

微分方程是 A-Level 数学和现实世界建模的核心内容，它们描述一个量相对于另一个量的变化方式。本文将重点讨论形如 dy/dx = f(x) 的一阶微分方程，并将相同思想推广到 dy/dx = g(y) 及可分离变量的相关形式。我们将涵盖通解、特解、应用和数值方法。

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

A differential equation is an equation that involves an unknown function and its derivatives. For example, dy/dx = 2x is a simple first-order differential equation, where y is the dependent variable and x is the independent variable. The order of a differential equation is the highest derivative present; here it is first order.

微分方程是含有未知函数及其导数的方程。例如，dy/dx = 2x 是一个简单的一阶微分方程，其中 y 是因变量，x 是自变量。微分方程的阶是方程中最高导数的阶数；这里的方程是一阶的。

In AQA A-Level Mathematics, you are expected to recognise and solve certain standard forms. The simplest form is dy/dx = f(x), where the derivative is given explicitly as a function of x. This form appears frequently in kinematics, growth processes and geometry.

在 AQA A-Level 数学中，你需要识别并求解某些标准形式。最简单的形式是 dy/dx = f(x) ，即导数被显式地表示为 x 的函数。这种形式在运动学、增长过程和几何问题中经常出现。

2. The Standard Form dy/dx = f(x) | 标准形式 dy/dx = f(x)

The equation dy/dx = f(x) states that the gradient of the curve y at any point x is determined solely by x. To find y, we reverse differentiation, i.e. integrate both sides with respect to x.

方程 dy/dx = f(x) 表示曲线 y 在任意点 x 处的斜率仅由 x 决定。要求 y，我们需要进行微分运算的逆运算，即对两边关于 x 积分。

y = ∫ f(x) dx + C

The constant C appears because integration is the reverse of differentiation and the derivative of any constant is zero. This expression is called the general solution of the differential equation.

积分后会出现常数 C，因为积分是微分的逆运算，而任意常数的导数都是零。这个表达式称为微分方程的通解。

3. Direct Integration: The General Solution | 直接积分：通解

To solve dy/dx = f(x), simply integrate the right-hand side term by term. For example, if dy/dx = 3x² + 2, then integrating gives y = x³ + 2x + C. Always include the arbitrary constant when no initial condition is given.

要解 dy/dx = f(x)，只需逐项对右边进行积分。例如，若 dy/dx = 3x² + 2，则积分得 y = x³ + 2x + C。当没有给出初始条件时，一定要包含任意常数 C。

∫ (3x² + 2) dx = x³ + 2x + C

In more formal terms, the general solution to dy/dx = f(x) is the family of curves that share the same gradient function. Each choice of C corresponds to a different vertical translation of the curve.

更正式地说，dy/dx = f(x)

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