📚 IB & WJEC Mathematics: Mastering Second-Order Differential Equations | IB WJEC 数学：二阶微分方程考点精讲
Second-order differential equations form a vital part of the IB and WJEC mathematics syllabuses, bridging pure calculus with real-world modelling. This article breaks down the essential theory, solution methods, and exam techniques you need to master, including the characteristic equation, three cases for homogeneous solutions, the method of undetermined coefficients for non-homogeneous equations, and applications of initial conditions.
二阶微分方程是 IB 和 WJEC 数学大纲中的核心内容，连接了纯微积分与现实建模。本文将逐一拆解你需要掌握的关键理论、求解方法和应考技巧，涵盖特征方程、齐次解三种情形、非齐次方程的待定系数法以及初始条件的应用。
1. What Is a Second-Order Differential Equation? | 什么是二阶微分方程？
A second-order differential equation involves an unknown function y(x), its first derivative dy/dx, and its second derivative d²y/dx². IB and WJEC papers focus on linear equations with constant coefficients, which appear in mechanics, electronics, and population models.
二阶微分方程包含未知函数 y(x)、它的一阶导数 dy/dx 和二阶导数 d²y/dx²。IB 和 WJEC 试题主要考查常系数线性方程，它们在力学、电子学和种群模型中随处可见。
2. General Form and Classification | 一般形式与分类
The general linear second-order ODE with constant coefficients can be written as:
常系数线性二阶常微分方程的一般形式可写作：
a d²y/dx² + b dy/dx + c y = f(x)
If f(x) = 0, the equation is homogeneous; otherwise, it is non-homogeneous. The constants a, b, and c are given, and usually a ≠ 0. Examiners expect you to identify the type immediately because the solution strategy depends on it.
如果 f(x) = 0，方程为齐次；否则为非齐次。常数 a, b, c 已知，通常 a ≠ 0。考官希望你立即识别方程类型，因为求解策略由此决定。
3. Homogeneous Equations and the Characteristic Equation | 齐次方程与特征方程
For the homogeneous case a d²y/dx² + b dy/dx + c y = 0, we assume a trial solution of the form y = e^(λx). Substituting this yields the characteristic equation:
对于齐次情形 a d²y/dx² + b dy/dx + c y = 0，我们假设试解 y = e^(λx)。代入后得到特征方程：
aλ² + bλ + c = 0
The nature of the roots λ₁ and λ₂ governs the structure of the general solution. Both IB and WJEC require you to solve this quadratic accurately and interpret the discriminant Δ = b² – 4ac.
根 λ₁ 和 λ₂ 的性质决定了通解的结构。IB 和 WJEC 都要求准确求解这个二次方程，并解释判别式 Δ = b² – 4ac。
4. Case 1: Two Distinct Real Roots | 情形1：两个不等实根
When Δ > 0, the characteristic equation gives two distinct real roots λ₁ and λ₂. The homogeneous general solution is a linear combination of two independent exponential functions:
当 Δ > 0 时，特征方程给出两个不等实根 λ₁ 和 λ₂。齐次通解是两个独立指数函数的线性组合：
y = A e^(λ₁x) + B e^(λ₂x)
A and B are arbitrary constants. This case is the most straightforward, and exam questions often use it to test substitution into initial conditions.
A 和 B 是任意常数。这是最直接的情形，考题常用它来检验代入初始条件的能力。
5. Case 2: Repeated Real Roots | 情形2：重实根
If Δ = 0, we obtain a double root λ = -b/(2a). Because two independent solutions are needed, the second solution gains a factor of x:
如果 Δ = 0，我们得到一个重根 λ = -b/(2a)。由于需要两个独立解，第二个解获得一个 x 因子：
y = (A + Bx) e^(λx)
Many students forget the ‘x’ multiplying B. Always double-check that the repeated root case is not mistaken for distinct roots.
很多同学会忘记 B 要乘以 x。务必反复检查，不要把重根情形误当作不等实根。
6. Case 3: Complex Conjugate Roots | 情形3：共轭复根
When Δ < 0, the roots are complex conjugates λ = α ± iβ. The real general solution uses sine and cosine, avoiding complex exponentials in the final answer:
当 Δ < 0 时，根为共轭复数 λ = α ± iβ。实数通解使用正弦和余弦，避免在最终答案中出现复数指数：
y = e^(αx)[C cos(βx) + D sin(βx)]
Here α = -b/(2a) and β = √(4ac – b²)/(2a). This form appears frequently in damped harmonic motion problems. IB and WJEC may ask you to explain the physical meaning of α (damping coefficient) and β (angular frequency).
这里 α = -b/(2a)，β = √(4ac – b²)/(2a)。这种形式在阻尼简谐运动问题中经常出现。IB 和 WJEC 可能会要求你解释 α（阻尼系数）和 β（角频率）的物理意义。
7. Constructing the General Solution for Homogeneous ODEs | 构造齐次常微分方程的通解
Once the roots are classified, you can write the complementary function yc. A summary table helps memorise the three patterns:
一旦完成了根的归类，你就可以写出余函数 yc。下面这个汇总表有助于记忆三种形式：
| Discriminant Δ | Roots | Homogeneous Solution yc |
|---|---|---|
| Δ > 0 | Real distinct λ₁, λ₂ | A e^(λ₁x) + B e^(λ₂x) |
| Δ = 0 | Repeated real λ | (A + Bx) e^(λx) |
| Δ < 0 | Complex α ± iβ | e^(αx)[A cos(βx) + B sin(βx)] |
Always state yc explicitly before moving to a particular integral.
在转向特解之前，一定要明确写出 yc。
8. Non-Homogeneous Equations: The Method of Undetermined Coefficients | 非齐次方程：待定系数法
For a non-homogeneous equation a d²y/dx² + b dy/dx + c y = f(x), the general solution is y = yc + yp, where yp is any particular integral. The method of undetermined coefficients works when f(x) is a polynomial, exponential, sine, cosine, or a sum of these.
对于非齐次方程 a d²y/dx² + b dy/dx + c y = f(x)，通解为 y = yc + yp，其中 yp 是任一特解。当 f(x) 为多项式、指数函数、正弦、余弦或它们的和时，就可以使用待定系数法。
You propose a trial yp with unknown coefficients that mirrors the form of f(x). Substitute into the ODE, equate coefficients, and solve for the constants. This method is directly tested in both IB and WJEC exams.
你需要根据 f(x) 的形式假设一个含有待定系数的试解 yp。将其代回原方程，比较系数并解出未知常数。IB 和 WJEC 考试都会直接考查这一方法。
9. Choosing the Correct Trial Particular Integral | 选择正确的试设特解形式
Matching the form of yp to f(x) is crucial. The rule: if any term in the trial yp already appears in the complementary function yc, multiply by x (or x²) until it is independent. Common pairings are:
让 yp 与 f(x) 匹配非常关键。规则是：若试设 yp 中的某项已在余函数 yc 中出现，则乘以 x（或 x²）直到独立。常见对应关系如下：
| f(x) | Trial yp (if no overlap with yc) |
|---|---|
| Polynomial of degree n | Anxⁿ + … + A₁x + A₀ |
| p e^(kx) | C e^(kx) |
| p cos(ωx) + q sin(ωx) | C cos(ωx) + D sin(ωx) |
| Product, e.g., x e^(kx) | (Cx + D) e^(kx) (adjust if yc contains e^(kx)) |
If the standard trial yp coincides with a term in yc, multiply the whole trial by x. For repeated overlaps, multiply by x². This adjustment is a frequent source of marks.
如果标准试设 yp 与 yc 中的项重合，就将整个试设乘以 x。若重复重合，则乘以 x²。这一调整是常见的得分点。
10. Applying Initial or Boundary Conditions | 应用初始条件或边界条件
Once the general solution y = yc + yp is found, you often need to determine the constants A and B using given conditions like y(0) = k and y'(0) = m. Differentiate the general solution, substitute the x-values, and solve the simultaneous equations.
求得通解 y = yc + yp 后，你通常需要利用给定的条件（如 y(0) = k 和 y'(0) = m）来确定常数 A 和 B。对通解求导，代入 x 值，然后求解联立方程组。
Be systematic: write y and dy/dx clearly, substitute, and simplify. Many marks are lost through algebraic slips when handling exponential or trigonometric derivatives.
要有条理：清晰写出 y 和 dy/dx，代值并化简。在处理指数或三角函数的导数时，许多失分源于代数运算中的疏忽。
11. Applications to Simple Harmonic Motion and Damping | 简谐运动与阻尼中的应用
Second-order ODEs model physical systems like a mass on a spring: m d²x/dt² + γ dx/dt + kx = F(t). In IB and WJEC, you might be asked to interpret the terms: m is mass, γ is damping constant, k is spring stiffness, and F(t) is external force.
二阶常微分方程可以模拟弹簧振子等物理系统：m d²x/dt² + γ dx/dt + kx = F(t)。在 IB 和 WJEC 中，你可能需要解释各项的含义：m 是质量，γ 是阻尼常数，k 是劲度系数，F(t) 是外力。
When F(t) = 0, the homogeneous solution describes free vibration. Complex roots give underdamped oscillation; repeated real roots give critical damping; distinct real roots give overdamped motion. Recognising these behaviours can earn valuable applied-marks.
当 F(t) = 0 时，齐次解描述了自由振动。复根对应欠阻尼振荡，重实根对应临界阻尼，不等实根对应过阻尼运动。能识别这些行为可以赢得宝贵的应用题分数。
12. Examination Tips and Common Pitfalls | 考试技巧与常见陷阱
Always identify the type of ODE before starting. Write down the characteristic equation clearly and factor or use the quadratic formula. Check the discriminant to avoid sign errors in the case of complex roots.
开始解题前，务必先判断方程的类型。清晰写出特征方程，采用因式分解或求根公式。检查判别式的符号，避免在处理复根时出现符号错误。
For undetermined coefficients, double-check that the trial yp is not already included in yc. If it is, multiply by x. Ensure that you have differentiated correctly when substituting into the original ODE, and that you have matched coefficients for both sides of the equation.
使用待定系数法时，要反复确认试设 yp 没有被包含在 yc 中。如果有，就乘以 x。确保代回原方程时求导正确，并且等号两边的系数已经一一对应。
Finally, practise with past IB and WJEC questions. Familiarity with the specific layout of marks—for finding the CF, selecting the PI, and applying conditions—will greatly improve your speed and accuracy in the exam.
最后，要多练习往年的 IB 和 WJEC 真题。熟悉得分点的具体分布——求解余函数、选择特解、应用条件——将极大地提高你在考试中的速度和准确性。
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