Mastering Second Order Differential Equations for OCR A-Level | OCR A-Level 二阶微分方程考点精讲

📚 Mastering Second Order Differential Equations for OCR A-Level | OCR A-Level 二阶微分方程考点精讲

A second-order differential equation lies at the heart of many physical and engineering problems, from the motion of a vibrating spring to the decay of an electric circuit. In OCR A-Level Mathematics (Further Pure), you are expected to solve linear second-order ordinary differential equations with constant coefficients, both homogeneous and non-homogeneous, and to apply initial or boundary conditions. This article distils the essential methods, common pitfalls, and exam-winning strategies into a clear, bilingual review.

二阶微分方程是许多物理和工程问题的核心，从弹簧振动到电路衰减。在 OCR A-Level 数学（Further Pure）中，你需要掌握求解常系数线性二阶常微分方程的方法，包括齐次和非齐次方程，并能运用初始条件或边界条件。本文将以清晰的中英双语形式，提炼核心解法、常见错误与应试策略。

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1. General Form and Key Terminology | 一般形式与核心术语

f(x) = 0, the equation is homogeneous; otherwise it is non-homogeneous. The solution y = y_c + y_p consists of the complementary function y_c (solving the homogeneous equation) and a particular integral y_p (any function satisfying the full equation).

常系数线性二阶常微分方程的形式为：a · d²y/dx² + b · dy/dx + c · y = f(x)。其中 a, b, c 为常数（a ≠ 0），f(x) 称为强制函数。当 f(x) = 0 时，方程为齐次方程，否则为非齐次方程。解的结构为 y = y_c + y_p，其中 y_c 是余函数（齐次方程的通解），y_p 是特解（满足非齐次方程的任意函数）。

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2. The Homogeneous Equation and Auxiliary Equation | 齐次方程与辅助方程

To find y_c, set f(x) = 0 and assume a trial solution y = e^mx. Differentiating gives dy/dx = m e^mx and d²y/dx² = m² e^mx s m₁ and m₂ of this quadratic completely determine the form of the complementary function.

求 y_c 时，令 f(x) = 0，并设试探解 y = e^mx。求导得 dy/dx = m e^mx，d²y/dx² = m² e^mx。代入齐次方程即得辅助方程：a m² + b m + c = 0。该二次方程的根 m₁ 和 m₂ 完全决定了余函数的形式。

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3. Distinct Real Roots | 相异实根

When the discriminant b² - 4ac > 0, the auxiliary equation has two distinct real roots m₁ and m₂. The complementary function is y_c = A e^{m₁ x} + B e^{m₂ x}, where A and B are arbitrary constants. This is the simplest case and appears frequently in exam questions involving exponential growth or decay dynamics.

当判别式 b² - 4ac > 0 时，辅助方程有两个相异实根 m₁ 和 m₂。余函数为 y_c = A e^{m₁ x} + B e^{m₂ x}，其中 A 与 B 为任意常数。这是最简单的情形，常见于涉及指数增长或衰减动态的考题。

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4. Repeated Real Root | 重实根

auxiliary equation has one repeated root m = -b/(2a). The complementary function is now y_c = (A + B x) e^{m x}. The extra factor of x is essential to capture the full set of independent solutions; forgetting this is one of the most frequent errors in exams.

若 b² - 4ac = 0，辅助方程有一个重根 m = -b/(2a)。此时余函数为 y_c = (A + B x) e^{m x}。额外的 x 因子是获得完整独立解组的关键；考试中忘记乘上 x 是最常见的错误之一。

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5. Complex Conjugate Roots | 共轭复根

When b² - 4ac < 0, the roots are complex conjugates m = α ± iβ, where α = -b/(2a) and β = √4ac - b²/(2a). The complementary function takes the oscillatory form y_c = e^{α x} (A cos β x + B sin β x). This case models damped vibrations, and the constants A and B are typically determined by initial conditions such as initial displacement and velocity.

当 b² - 4ac < 0 时，根为一对共轭复数 m = α ± iβ，其中 α = -b/(2a)，β = √4ac - b²/(2a)。余函数呈振荡形式：y_c = e^{α x} (A cos β x + B sin β x)。这类情况用于建模阻尼振动，常数 A、B 通常由初始位移和初始速度等条件确定。

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6. Non-Homogeneous Equations: Choosing a Particular Integral | 非齐次方程：特解的选定

For a non-homogeneous equation a d²y/dx² + b dy/dx + c y = f(x), the particular integral y_p is found by the method of undetermined coefficients. The first step is to guess a form for y_p based on the type of f(x). The guess must not duplicate any term already present in y_c by x (or x²) until all terms are distinct. Always write the full general solution as y = y_c + y_p before substituting conditions.

对于非齐次方程 a d²y/dx² + b dy/dx + c y = f(x)，特解 y_p 通过待定系数法求得。首先根据 f(x) 的类型猜测一个 y_p 的形式。该形式不得与 y_c 中的任何项重复；若出现重复，则需将猜测项乘以 x（或 x²）直至所有项互异。在代入条件求常数前，务必写出完整的通解 y = y_c + y_p。

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7. Standard Guesses for f(x) | f(x) 的标准猜测形式

The table below summarises the standard trial functions for common forcing terms. Always use the full polynomial even if some terms are missing; for trigonometric functions, include both sine and cosine even when only one appears in f(x). Remember: if the standard guess overlaps with y_c, multiply by x and re-check.

下表总结了常见强制项的标准试探函数。即使多项式缺少某些项，也应使用完整的同次多项式；对于三角函数，即使 f(x) 中只出现正弦或余弦，试探解也需同时包含正弦和余弦项。切记：若标准猜测与 y_c 重叠，则乘以 x 后重新检验。

f(x)	Trial yp
constant k	C
polynomial Pₙ(x) of degree n	Aₙ xⁿ + Aₙ₋₁ xⁿ⁻¹ + ... + A₀
k ep x	C ep x (unless p is a root)
k sin ωx or k cos ωx	C cos ωx + D sin ωx
k x ep x etc.	(C x + D) ep x if p is a repeated root of auxiliary eqn

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8. Modifying Guesses When Overlap Occurs | 重叠时的猜测修正

If the unmodified trial y_p shares a term with y_c, multiply the entire trial y_p by x (or by x² if the overlap persists). For instance, if y_c contains e^2x and f(x) = 5 e^2x, the initial guess y_p = C e^2x is invalid. Multiply by x to get y_p = C x e^2x. If 2 is a repeated root, use y_p = C x² e^2x. This rule avoids missing a valid particular integral.

若未经修正的试探 y_p 与 y_c 有共同项，则将整个试探 y_p 乘以 x（若仍重叠则进一步乘以 x²）。例如 y_c 含有 e^2x 而 f(x) = 5 e^2x，初始猜测 y_p = C e^2x 无效，需乘以 x 得 y_p = C x e^2x。若 2 为重根，则采用 y_p = C x² e^2x。此规则可避免遗漏有效的特解。

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9. Substituting and Solving for Coefficients | 代入并求解系数

Once the trial y_p is chosen, compute its first and second derivatives, substitute them into the original non-homogeneous ODE, and equate coefficients of like terms. This yields a system of linear equations for the unknown constants. Solve them carefully; algebraic slips are common. After finding y_p, write y = y_c + y_p and proceed to impose any given conditions.

选定试探 y_p 后，求出它的一阶和二阶导数，代入原非齐次常微分方程，并比较同类项系数，从而得到关于未知常数的线性方程组。请仔细求解，代数失误是常见丢分点。求出 y_p 后，写出 y = y_c + y_p，然后代入所给条件。

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10. Using Initial or Boundary Conditions | 运用初始条件或边界条件

olution and its derivative to obtain two equations in the arbitrary constants (A, B from y_c). Solve simultaneously to find the unique particular solution. Always differentiate y before plugging in conditions — and double-check the arithmetic.

OCR 考题通常会给出两个条件，例如当 x = 0 时 y = 1 且 dy/dx = 0。将 x = 0 代入通解及其导数，得到关于任意常数（y_c 中的 A、B）的两个方程，联立求解即得唯一特解。务必先将 y 求导再代入条件，并核验每一步计算。

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11. Applications: SHM and Damped Oscillations | 应用：简谐运动与阻尼振动

bes a damped harmonic oscillator. For underdamping (complex roots), the displacement x = e^{α t} (A cos β t + B sin β t) exhibits decaying oscillations — the real part α controls the envelope. These models appear regularly in mechanics and circuit questions.

一个经典应用是运动方程 m d²x/dt² + λ dx/dt + k x = F(t)。当无外力时，齐次方程描述阻尼谐振子。在欠阻尼（复根）情形下，位移 x = e^{α t} (A cos β t + B sin β t) 呈现衰减振荡，实部 α 控制包络线。这类模型频繁出现在力学与电路考题中。

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12. Exam Strategy and Common Pitfalls | 应试策略与常见陷阱

Always begin by identifying a, b, c and the form of f(x). Write the auxiliary equation clearly and state its discriminant to avoid sign errors. When guessing y_p, check for overlap with y_c before solving. Keep constants A and B until the very end; premature substitution loses generality. Finally, leave your final answer in the requested form, and under- line the key steps to gain method marks.

解题时，首先识别 a、b、c 及 f(x) 的形式。清晰写出辅助方程并注明显判别式以防符号错误。猜测 y_p 时，务必先检验是否与 y_c 重叠。将常数 A、B 保留到最后，过早代入会丧失一般性。最后按照题目要求的形式给出答案，并突出关键步骤以获得过程分。

Published by TutorHao | Further Pure Mathematics Revision Series | aleveler.com

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