Second Order Differential Equations for AQA | A-Level AQA 数学：二阶微分方程 考点精讲

📚 Second Order Differential Equations for AQA | A-Level AQA 数学：二阶微分方程 考点精讲

Second order differential equations form a cornerstone of the AQA A-Level Mathematics syllabus, bridging pure calculus with powerful applications in mechanics and physics. These equations involve the second derivative of a function and allow us to model dynamic systems such as oscillating springs, damped vibrations, and electrical circuits. Mastering this topic requires a firm grasp of auxiliary equations, complementary functions, particular integrals, and the physical interpretation of boundary conditions.

二阶微分方程是 AQA A-Level 数学考纲的核心内容，将纯微积分与力学和物理学中的强大应用连接起来。这些方程涉及函数的二阶导数，使我们能够对动态系统进行建模，例如弹簧振动、阻尼振动和电路。掌握这一主题需要牢固掌握辅助方程、余函数、特积分以及边界条件的物理解释。

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1. Introduction to Second Order ODEs | 二阶常微分方程简介

A second order ordinary differential equation (ODE) involves an unknown function y(x) and its derivatives up to the second order. In the AQA specification, we focus on linear ODEs with constant coefficients of the form a d²y/dx² + b dy/dx + cy = f(x), where a, b, and c are constants and f(x) is a function of x. The order is determined by the highest derivative present — in this case, the second derivative d²y/dx².

二阶常微分方程 (ODE) 包含未知函数 y(x) 及其直到二阶的导数。在 AQA 考纲中，我们专注于常系数线性常微分方程，形式为 a d²y/dx² + b dy/dx + cy = f(x)，其中 a、b 和 c 是常数，f(x) 是 x 的函数。阶数由出现的最高阶导数决定 —— 在这种情况下，是二阶导数 d²y/dx²。

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2. Homogeneous vs Non-Homogeneous Equations | 齐次方程与非齐次方程

When the right-hand side f(x) equals zero, the equation a d²y/dx² + b dy/dx + cy = 0 is called homogeneous. The solution to this homogeneous equation is the complementary function (CF), denoted y_c. When f(x) is not zero, we must find a particular integral (PI), denoted y_p, which satisfies the full equation. The general solution is the sum y = y_c + y_p. This superposition principle arises because the ODE is linear.

当右边 f(x) 等于零时，方程 a d²y/dx² + b dy/dx + cy = 0 称为齐次方程。这个齐次方程的解是余函数 (CF)，记作 y_c。当 f(x) 不为零时，我们必须找到一个特积分 (PI)，记作 y_p，它满足完整的方程。通解是 y = y_c + y_p 的和。这个叠加原理成立是因为常微分方程是线性的。

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3. The Auxiliary Equation Method | 辅助方程法

To solve the homogeneous equation a d²y/dx² + b dy/dx + cy = 0, we assume a trial solution of the form y = e^λx. Substituting this into the ODE yields the auxiliary equation aλ² + bλ + c = 0. This is a quadratic equation in λ, and its roots determine the form of the complementary function. The process transforms differentiation into algebraic multiplication, making the solution systematic and elegant.

要求解齐次方程 a d²y/dx² + b dy/dx + cy = 0，我们假设一个试探解 y = e^λx。将其代入常微分方程得到辅助方程 aλ² + bλ + c = 0。这是一个关于 λ 的二次方程，其根决定了余函数的形式。这个过程将微分转化为代数乘法，使求解变得系统而优雅。

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4. Real Distinct Roots | 实不相等的根

If the auxiliary equation has two distinct real roots λ₁ and λ₂, the complementary function is y_c = Ae^λ₁x + Be^λ₂x, where A and B are arbitrary constants. This case arises when the discriminant b² − 4ac > 0. For example, solving d²y/dx² − 5 dy/dx + 6y = 0 gives the auxiliary equation λ² − 5λ + 6 = 0, with roots λ = 2 and λ = 3, so y_c = Ae^2x + Be^3x.

如果辅助方程有两个不同的实根 λ₁ 和 λ₂，余函数为 y_c = Ae^λ₁x + Be^λ₂x，其中 A 和 B 是任意常数。这种情况发生在判别式 b² − 4ac > 0 时。例如，求解 d²y/dx² − 5 dy/dx + 6y = 0 得到辅助方程 λ² − 5λ + 6 = 0，其根为 λ = 2 和 λ = 3，因此 y_c = Ae^2x + Be^3x。

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5. Repeated Roots | 重根

When the discriminant equals zero, the auxiliary equation has a repeated root λ (multiplicity 2). In this case, the complementary function takes the form y_c = (A + Bx)e^λx. The extra factor of x ensures linear independence of the two solutions. For instance, d²y/dx² − 4 dy/dx + 4y = 0 leads to λ² − 4λ + 4 = 0 with a repeated root λ = 2, giving y_c = (A + Bx)e^2x.

当判别式等于零时，辅助方程有一个重根 λ（重数为 2）。在这种情况下，余函数的形式为 y_c = (A + Bx)e^λx。额外的因子 x 确保了两个解的线性无关性。例如，d²y/dx² − 4 dy/dx + 4y = 0 导出 λ² − 4λ + 4 = 0，具有重根 λ = 2，得到 y_c = (A + Bx)e^2x。

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6. Complex Conjugate Roots | 共轭复根

If the discriminant is negative, the auxiliary equation yields complex conjugate roots λ = α ± iβ, where i is the imaginary unit. The complementary function is then y_c = e^αx(A cos βx + B sin βx). This form is crucial for modelling oscillatory motion without damping. For example, d²y/dx² + 4y = 0 gives λ² + 4 = 0, so λ = ±2i, leading to y_c = A cos 2x + B sin 2x. Here α = 0 and β = 2.

如果判别式为负，辅助方程产生共轭复根 λ = α ± iβ，其中 i 是虚数单位。余函数则为 y_c = e^αx(A cos βx + B sin βx)。这种形式对于无阻尼的振荡运动建模至关重要。例如，d²y/dx² + 4y = 0 给出 λ² + 4 = 0，所以 λ = ±2i，得到 y_c = A cos 2x + B sin 2x。这里 α = 0，β = 2。

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7. Particular Integrals: The Method of Undetermined Coefficients | 特积分：待定系数法

For a non-homogeneous equation a d²y/dx² + b dy/dx + cy = f(x), we seek a particular integral y_p that mirrors the form of f(x). We guess a trial function with undetermined coefficients, substitute into the ODE, and equate coefficients to find the constants. The standard trial forms are:

• f(x) = polynomial of degree n → y_p = general polynomial of degree n
• f(x) = ke^px → y_p = Ce^px
• f(x) = m cos ωx + n sin ωx → y_p = P cos ωx + Q sin ωx

对于非齐次方程 a d²y/dx² + b dy/dx + cy = f(x)，我们寻找一个与 f(x) 形式相似的特积分 y_p。我们猜测一个含有待定系数的试探函数，代入常微分方程，并比较系数以求出常数。标准的试探形式为：

• f(x) = n 次多项式 → y_p = n 次一般多项式
• f(x) = ke^px → y_p = Ce^px
• f(x) = m cos ωx + n sin ωx → y_p = P cos ωx + Q sin ωx

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8. Handling Failures of the Trial Function | 处理试探函数失效的情况

If the trial function already appears in the complementary function (a case of resonance), we must multiply the trial y_p by x (or by x² if the root is repeated) to achieve linear independence. For example, if f(x) = e^2x and the CF already contains e^2x, the trial becomes y_p = Cxe^2x. Similarly, for trigonometric f(x) that matches the frequency of the CF, we multiply by x.

如果试探函数已经出现在余函数中（共振情况），我们必须将试探 y_p 乘以 x（如果根是重根则乘以 x²）以实现线性无关。例如，如果 f(x) = e^2x 而 CF 已经包含 e^2x，试探变为 y_p = Cxe^2x。类似地，对于与 CF 频率匹配的三角 f(x)，我们乘以 x。

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9. Boundary and Initial Conditions | 边界条件与初始条件

After obtaining the general solution y = y_c + y_p with arbitrary constants, we use given conditions to find particular solutions. For second order ODEs, we need two conditions. These may be boundary conditions (values of y at two different x-values) or initial conditions (values of y and dy/dx at a single x, often x = 0). Substituting the conditions yields simultaneous equations to determine the constants A and B.

在得到带有任意常数的通解 y = y_c + y_p 之后，我们使用给定的条件来求特解。对于二阶常微分方程，我们需要两个条件。这些可以是边界条件（在两个不同 x 值处的 y 值）或初始条件（在单个 x 处，通常 x = 0 时 y 和 dy/dx 的值）。代入条件会得到联立方程，用于确定常数 A 和 B。

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10. Modelling Simple Harmonic Motion | 建模简谐运动

One of the most important applications in AQA Mechanics is simple harmonic motion (SHM), governed by d²x/dt² = −ω²x, or equivalently d²x/dt² + ω²x = 0. The auxiliary equation λ² + ω² = 0 gives λ = ±iω, so the general solution is x = A cos ωt + B sin ωt. This can be rewritten as x = R cos(ωt − φ) or x = R sin(ωt + φ), where R is the amplitude and φ is the phase angle. The period is T = 2π/ω.

在 AQA 力学中最重要的应用之一是简谐运动 (SHM)，其方程为 d²x/dt² = −ω²x，或等价地 d²x/dt² + ω²x = 0。辅助方程 λ² + ω² = 0 给出 λ = ±iω，因此通解为 x = A cos ωt + B sin ωt。这可以重写为 x = R cos(ωt − φ) 或 x = R sin(ωt + φ)，其中 R 是振幅，φ 是相位角。周期为 T = 2π/ω。

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11. Damped Harmonic Motion | 阻尼振动分析

When a damping force proportional to velocity is added, the equation becomes a d²x/dt² + b dx/dt + cx = 0. The auxiliary equation aλ² + bλ + c = 0 leads to three regimes:

• Overdamping (b² > 4ac): distinct real roots, non-oscillatory decay
• Critical damping (b² = 4ac): repeated real root, fastest non-oscillatory decay
• Underdamping (b² < 4ac): complex roots, oscillatory decay with envelope e^−bt/(2a)

The AQA exam often asks students to identify which regime applies and sketch the corresponding motion graph.

当添加与速度成正比的阻尼力时，方程变为 a d²x/dt² + b dx/dt + cx = 0。辅助方程 aλ² + bλ + c = 0 导出三种情况：

• 过阻尼 (b² > 4ac)：不同实根，非振荡衰减
• 临界阻尼 (b² = 4ac)：重实根，最快的非振荡衰减
• 欠阻尼 (b² < 4ac)：复根，带有包络线 e^−bt/(2a) 的振荡衰减

AQA 考试常要求学生识别适用哪种情况并绘制相应的运动图形。

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12. Common Pitfalls and Exam Strategy | 常见错误与考试策略

Students frequently confuse the sign when writing the auxiliary equation — ensure that the signs from the ODE are copied exactly. Another common error is forgetting to multiply the trial function by x when overlap with the CF occurs. In exam scenarios, always write the general solution clearly before substituting conditions, and check that your particular integral actually satisfies the original ODE. When sketching damped oscillations, label the exponential envelope and key intercepts accurately.

学生在写辅助方程时常常把符号弄混 —— 确保从常微分方程中完全照抄符号。另一个常见错误是当试探函数与 CF 重叠时忘记将其乘以 x。在考试情境中，务必在代入条件之前清晰地写出通解，并检查你的特积分是否真的满足原方程。在绘制阻尼振荡时，准确标记指数包络线和关键的截距。

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