Second Order Differential Equations | 二阶微分方程考点精讲

📚 Second Order Differential Equations | 二阶微分方程考点精讲

Second order differential equations are a core topic in the WJEC IGCSE Further Mathematics syllabus. They extend the ideas from first order equations to situations where the rate of change itself is changing. In this revision guide, we will cover the key methods you need to solve homogeneous and non‑homogeneous linear second order ODEs with constant coefficients, interpret their solutions, and apply them to real‑world problems. Every step is designed to build your confidence for the exam.

二阶微分方程是 WJEC IGCSE 进阶数学大纲中的核心内容。它将一阶方程的思想推广到变化率本身也在变化的情形。本考点精讲将覆盖你需要掌握的解题方法，包括常系数线性二阶齐次与非齐次方程，解的解释，以及实际应用。每一步都旨在大幅提升你应对考试的信心。

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1. Introduction to Second Order Differential Equations | 二阶微分方程简介

A differential equation is an equation involving an unknown function and its derivatives. A second order differential equation contains the second derivative d²y/dx². In IGCSE, we focus on linear equations where y and its derivatives appear only to the first power and are not multiplied together. These equations model acceleration in mechanics, oscillations in springs, and charge in electrical circuits.

微分方程是包含未知函数及其导数的方程。二阶微分方程含有二阶导数 d²y/dx²。在 IGCSE 中，我们关注线性方程，其中 y 及其导数仅以一次幂出现且彼此不乘积。此类方程常用来描述力学中的加速度、弹簧振动和电路中的电荷变化。

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2. General Form of Second Order Linear ODE | 二阶线性常微分方程的一般形式

The most common form we work with is a d²y/dx² + b dy/dx + c y = f(x), where a, b, c are constants and a ≠ 0. If f(x) = 0, the equation is homogeneous; otherwise it is non‑homogeneous. The solution y = y_c + y_p combines the complementary function (CF) from the homogeneous part and a particular integral (PI) for the forcing term f(x).

我们最常处理的形式是 a d²y/dx² + b dy/dx + c y = f(x)，其中 a, b, c 为常数且 a ≠ 0。若 f(x) = 0，方程为齐次；否则为非齐次。通解 y = y_c + y_p，它由齐次部分的余函数 (CF) 和对应于驱动项 f(x) 的特解 (PI) 组合而成。

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3. Homogeneous Equations with Constant Coefficients | 常系数齐次方程

For a homogeneous equation a d²y/dx² + b dy/dx + c y = 0, we assume a solution of the form y = e^mx. Substituting this into the ODE gives the auxiliary equation a m² + b m + c = 0. The nature of the roots of this quadratic determines the form of the complementary function.

对于齐次方程 a d²y/dx² + b dy/dx + c y = 0，我们假设解的形式为 y = e^mx。将其代入方程得到特征方程 a m² + b m + c = 0。这个二次方程根的性质决定了余函数的形式。

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4. Auxiliary Equation and Types of Roots | 特征方程与根的类型

The auxiliary equation a m² + b m + c = 0 is solved using the quadratic formula m = [−b ± √(b² − 4ac)] / (2a). The discriminant Δ = b² − 4ac leads to three cases: (i) Δ > 0 gives two distinct real roots; (ii) Δ = 0 gives a repeated real root; (iii) Δ < 0 gives a pair of complex conjugate roots.

特征方程 a m² + b m + c = 0 用求根公式 m = [−b ± √(b² − 4ac)] / (2a) 求解。判别式 Δ = b² − 4ac 导致三种情况：(i) Δ > 0 给出两个相异实根；(ii) Δ = 0 给出一个重实根；(iii) Δ < 0 给出一对共轭复根。

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5. Real and Distinct Roots | 实根且不等根

If the auxiliary equation has two distinct real roots m₁ and m₂, the complementary function is y_c = A e^m₁x + B e^m₂x, where A and B are arbitrary constants. This solution represents a combination of exponential growth or decay terms, often seen in overdamped motion.

若特征方程有两个相异实根 m₁ 和 m₂，余函数为 y_c = A e^m₁x + B e^m₂x，其中 A 和 B 为任意常数。该解表示指数增长或衰减项的组合，常见于过阻尼运动。

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6. Repeated Real Roots | 重实根

When the discriminant is zero, we have a repeated root m. The CF is y_c = (A + Bx) e^mx. The extra factor of x ensures we have two linearly independent solutions. Exam questions often require you to verify that x e^mx indeed satisfies the ODE.

当判别式为零时，有一个重根 m。余函数为 y_c = (A + Bx) e^mx。额外的 x 因子确保我们有两个线性无关的解。考试常要求你验证 x e^mx 确实满足方程。

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7. Complex Conjugate Roots | 共轭复根

For Δ < 0 the auxiliary equation gives roots m = α ± iβ. The complementary function is y_c = e^αx [C cos(βx) + D sin(βx)] where C and D are constants. You may also see the form y_c = A e^αx cos(βx − φ) using amplitude and phase. This solution models underdamped oscillations.

对于 Δ < 0，特征方程的根为 m = α ± iβ。余函数为 y_c = e^αx [C cos(βx) + D sin(βx)]，其中 C 和 D 为常数。你也可以使用振幅和相位形式 y_c = A e^αx cos(βx − φ)。该解描述欠阻尼振荡。

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8. Solving Initial Value Problems | 解初值问题

To find the particular solution satisfying given conditions, substitute x = x₀, y = y₀, and dy/dx = v₀ into the CF (and its derivative). Solve the simultaneous equations to determine A and B, or C and D. This gives a unique solution curve fitting the initial data.

要找出满足给定初始条件的特解，将 x = x₀, y = y₀ 和 dy/dx = v₀ 代入余函数及其导数。解联立方程求出 A 和 B，或 C 和 D。这便得到符合初始数据的唯一解曲线。

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9. Introduction to Non-Homogeneous Equations | 非齐次方程简介

A non‑homogeneous second order linear ODE has the form a d²y/dx² + b dy/dx + c y = f(x) with f(x) ≠ 0. Its general solution is y = y_c + y_p, where y_c is the CF from the corresponding homogeneous equation, and y_p is any particular integral. The method of undetermined coefficients is used to find y_p when f(x) is a polynomial, exponential, sine/cosine, or a sum of such functions.

非齐次二阶线性常微分方程形如 a d²y/dx² + b dy/dx + c y = f(x)，且 f(x) ≠ 0。其通解为 y = y_c + y_p，其中 y_c 是对应齐次方程的余函数，y_p 是任一特解。当 f(x) 为多项式、指数函数、正弦/余弦或这些函数的和时，我们使用待定系数法求 y_p。

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10. Particular Integral for Polynomial/Exponential/Trigonometric | 特解（多项式/指数/三角）

To find y_p, we guess a function of the same form as f(x) with unknown coefficients, then substitute into the left‑hand side and equate coefficients. For f(x) = k (constant), try y_p = P. For f(x) = polynomial of degree n, try a general polynomial of degree n. For f(x) = p e^qx, try y_p = Q e^qx, unless q is a root of the auxiliary equation. For f(x) = p sin(rx) or q cos(rx), try y_p = M sin(rx) + N cos(rx). If the guess overlaps with the CF, multiply by x.

为求 y_p，我们猜测一个与 f(x) 同形的、含待定系数的函数，代入方程左端并比较系数。若 f(x) = k（常数），试设 y_p = P。若 f(x) 为 n 次多项式，试设一个 n 次一般多项式。若 f(x) = p e^qx，试设 y_p = Q e^qx，除非 q 是特征方程的根。若 f(x) = p sin(rx) 或 q cos(rx)，试设 y_p = M sin(rx) + N cos(rx)。如果猜测形式与 CF 重叠，则乘以 x。

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11. General Solution of Non-Homogeneous Equations | 非齐次方程的通解

Once y_c and y_p are found, the general solution is y = y_c + y_p. After writing the full solution, initial or boundary conditions can be applied to find the arbitrary constants. Remember that the constants come only from the CF part; the PI does not contain arbitrary constants.

一经求出 y_c 和 y_p，通解即为 y = y_c + y_p。写出完整解后，可应用初始条件或边界条件求出任意常数。注意常数仅来自 CF 部分；PI 不含任意常数。

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12. Applications in Physics and Contextual Problems | 物理背景应用题

WJEC questions often embed differential equations in contexts like simple harmonic motion d²x/dt² + ω²x = 0, damped vibrations, or electrical LCR circuits. Identify the homogeneous part, write down the auxiliary equation, read off the type of damping from the discriminant, and interpret the solution’s long‑term behaviour. In exponential growth/decay models, the sign of the real part α in complex roots determines stability.

WJEC 试题常将微分方程融入简谐运动 d²x/dt² + ω²x = 0、阻尼振动或 LCR 电路等情境。识别齐次部分，写出特征方程，根据判别式判断阻尼类型，并解释解的长期行为。在指数增长/衰减模型中，复根实部 α 的符号决定稳定性。

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