Second Order Differential Equations | 二阶微分方程

📚 Second Order Differential Equations | 二阶微分方程

A second order differential equation involves the second derivative d²y/dx² and is a key topic in AQA A-Level Mathematics. Mastering these equations allows you to model oscillations, electrical circuits and mechanical vibrations, and to find explicit solutions using clear algebraic methods.

二阶微分方程涉及二阶导数 d²y/dx²，是 AQA A-Level 数学的重要考点。掌握这类方程可以帮助你建立振动、电路和机械运动的模型，并通过清晰的代数方法求出显式解。

1. General Form and Terminology | 方程的一般形式与术语

A linear second order differential equation with constant coefficients has the form

常系数线性二阶微分方程的一般形式为

a d²y/dx² + b dy/dx + c y = f(x)

where a, b and c are constants and a ≠ 0. If f(x) = 0, the equation is homogeneous; otherwise it is non-homogeneous.

其中 a、b、c 为常数且 a ≠ 0。如果 f(x)=0，方程为齐次方程；否则为非齐次方程。

The solution usually has two parts: a complementary function (CF) from the homogeneous equation and a particular integral (PI) depending on f(x).

解通常包含两部分：由齐次方程得到的余函数（CF），以及由 f(x) 决定的特解（PI）。

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