Second-Order Differential Equations for AS Mathematics | AS 数学：二阶微分方程考点精讲

📚 Second-Order Differential Equations for AS Mathematics | AS 数学：二阶微分方程考点精讲

Mastering second-order differential equations is a crucial step in AS Mathematics, enabling you to solve problems involving oscillations, forces, and rates of change. This revision guide covers the essential techniques: solving homogeneous equations via the characteristic equation, classifying roots, forming the complementary function, and handling inhomogeneous cases with the method of undetermined coefficients, as well as applying boundary conditions to find particular solutions.

掌握二阶微分方程是AS数学的关键一步，使你能够解决涉及振动、力和变化率的问题。本复习指南涵盖必备技巧：通过特征方程求解齐次方程、根的判别、构建补函数、用待定系数法处理非齐次情形，以及应用边界条件求特解。

1. What Are Second-Order Differential Equations? | 什么是二阶微分方程？

A second-order differential equation relates a function y(x) with its first and second derivatives, y'(x) and y”(x). They naturally arise in physics when describing acceleration (which is the second derivative of position) or in simple harmonic motion. In AS Mathematics, we consider linear equations with constant coefficients: a d²y/dx² + b dy/dx + c y = f(x).

二阶微分方程将函数 y(x) 与其一阶和二阶导数 y'(x) 和 y”(x) 联系起来。它们在物理中描述加速度（位移的二阶导数）或简谐运动时自然出现。在AS数学中，我们重点学习常系数线性方程：a d²y/dx² + b dy/dx + c y = f(x)。

The solution consists of two parts: the complementary function (CF) obtained from the homogeneous equation (f(x)=0) and the particular integral (PI) for the inhomogeneous equation. The general solution is y = y_c + y_p. Understanding this structure is the foundation for all methods that follow.

解由两部分构成：从齐次方程 (f(x)=0) 得到的补函数（CF）和非齐次方程的特解（PI）。通解为 y = y_c + y_p。理解这一结构是所有后续方法的基础。

2. The Standard Form of a Linear Constant-Coefficient ODE | 常系数线性常微分方程的标准形式

We always write the equation in the form a y” + b y’ + c y = f(x) where a, b, c are constants and a ≠ 0. The term f(x) is called the forcing function. If f(x) = 0, the equation is homogeneous; otherwise, it is inhomogeneous (or non-homogeneous).

我们总是将方程写成 a y” + b y’ + c y = f(x) 的形式，其中 a, b, c 为常数且 a ≠ 0。项 f(x) 称为强迫函数。若 f(x) = 0，方程是齐次的；否则为非齐次。

For AS

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