📚 Unforced Linear Differential Equations with Constant Coefficients | 常系数齐次线性微分方程
In the IB Mathematics curriculum, differential equations serve as a bridge between abstract calculus and real-world modelling. Unforced linear differential equations with constant coefficients – often called homogeneous linear differential equations – form a cornerstone of this topic. They describe systems where no external force or input acts on the system, and understanding their solution structure is essential for progress in calculus, physics, and engineering.
在 IB 数学课程中，微分方程是连接抽象微积分与现实建模的桥梁。无外力线性常系数微分方程——通常称为齐次线性微分方程——构成了这一主题的基石。它们描述了没有外力或外部输入作用的系统，理解其解的结构对于微积分、物理和工程学的深入至关重要。
1. What Are Differential Equations? | 什么是微分方程？
A differential equation is any equation that relates a function with one or more of its derivatives. The order of a differential equation is the highest derivative that appears. Linear equations have derivatives appearing only to the first power and not multiplied together. 'Constant coefficients' means that the coefficients of the function and its derivatives are constants. Finally, 'unforced' or 'homogeneous' means that the equation is set to zero – no external forcing term is present.
微分方程是任意将函数与其一个或多个导数联系起来的方程。方程的阶数是指出现的最高阶导数。线性方程是指导数仅以一次幂出现且彼此不乘积。“常系数”指的是函数及其导数的系数均为常数。而“无外力”或“齐次”意味着方程右侧为零——不存在外加驱动项。
2. Second-Order Homogeneous Equations with Constant Coefficients | 二阶常系数线性齐次方程
e real constants and a ≠ 0. This equation appears in countless physical contexts, such as spring-mass systems without damping or simple electrical circuits.
二阶常系数无外力线性微分方程的标准形式为 a d²y/dx² + b dy/dx + c y = 0，其中 a、b、c 为实常数且 a ≠ 0。该方程出现在无数物理情境中，例如无阻尼的弹簧质量系统或简单的电路系统。
3. The Characteristic Equation | 特征方程
To solve the equation, we assume a trial solution of the form y = erx. Substituting this into the differential equation yields a r² erx + b r erx + c erx = 0. Since erx equation): a r² + b r + c = 0. The roots of this quadratic determine the general solution.
为了求解该方程，我们假设一个试解形式 y = erx。将其代入微分方程得到 a r² erx + b r erx + c erx = 0。由于 erx 永不为零，可约去该因子，得到特征方程（或辅助方程）：a r² + b r + c = 0。这个二次方程的根决定了通解的形式。
4. Case 1: Distinct Real Roots | 情况一：相异实根
eal roots r₁ and r₂, then the general solution of the differential equation is y(x) = C₁ er₁ x + C₂ er₂ x, where C₁ and C₂ are arbitrary constants. The functions er₁ x and er₂ x are linearly independent, which guarantees the completeness of the solution.
若特征方程有两个相异实根 r₁ 和 r₂，则微分方程的通解为 y(x) = C₁ er₁ x + C₂ er₂ x，其中 C₁ 和 C₂ 为任意常数。函数 er₁ x 与 er₂ x 线性无关，这就保证了解集的完整性。
5. Case 2: Repeated Real Root | 情况二：重实根
When the discriminant is zero, the characteristic equation has one repeated real root r. In this case, the two linearly independent solutions are er x and x er x. The general solution becomes y(x) = (C₁ + C₂ x) er x. It is important not to forget the factor x in the second term, as it is crucial for linear independence.
当判别式为零时，特征方程有一个重实根 r。此时，两个线性无关的解为 er x 和 x er x。通解变为 y(x) = (C₁ + C₂ x) er x。切记不要遗漏第二项中的因子 x，因为它对于保证线性无关至关重要。
6. Case 3: Complex Conjugate Roots | 情况三：共轭复根
written as r = α ± iβ, then the general solution can be expressed in real form as y(x) = eα x (C₁ cos βx + C₂ sin βx). Euler's formula, eiθ from complex exponentials to trigonometric functions. This form is particularly useful for modelling oscillatory behaviour.
如果根为共轭复数，记作 r = α ± iβ，则通解可以用实函数表示为 y(x) = eα x (C₁ cos βx + C₂ sin βx)。欧拉公式 eiθ = cos θ + i sin θ 使得我们能够将复指数转化为三角函数。这种形式对于描述振荡行为特别有用。
7. Applying Initial Conditions | 应用初始条件
To determine the particular solution, we use initial conditions such as y(0) and y'(0). By substituting these into the general solution and its derivative, we obtain a system of linear equations for C₁ and C₂. Solving this system yields the specific constants that satisfy the given starting state of the system.
为了确定特解，我们需要使用初始条件，例如 y(0) 和 y'(0)。将它们代入通解及其导数，即可得到关于 C₁ 和 C₂ 的线性方程组。求解此方程组便能得出满足系统初始状态的具体常数。
8. Higher-Order Homogeneous Equations | 高阶齐次方程
to nth-order linear homogeneous equations with constant coefficients: aₙ y⁽ⁿ⁾ + ... + a₁ y' ₀ = 0. For each real root r of multiplicity m, we contribute terms (C₁ + C₂ x + ... + Cₘ xm-1) er x. For complex conjugate pairs of multiplicity 1, we use eα x (C₁ cos βx + C₂ sin βx); higher multiplicities involve polynomial factors in a similar way.
该方法可自然推广到 n 阶常系数线性齐次方程：aₙ y⁽ⁿ⁾ + ... + a₁ y' + a₀ y = 0。其特征方程为 aₙ rⁿ + ... + a₁ r + a₀ = 0。对于每个 m 重实根 r，我们都贡献形式为 (C₁ + C₂ x + ... + Cₘ xm-1) er x 的项。对于一重共轭复根，我们使用 eα x (C₁ cos βx + C₂ sin βx)；更高重数的情况则以类似的方式引入多项式因子。
9. The Role of the Discriminant | 判别式的作用
or Δ < 0 we obtain complex conjugate roots. Summarising this relationship in a table can help when tackling exam problems quickly.
判别式 Δ = b² - 4ac 可以对三种根的情况进行快速分类。Δ > 0 时有两相异实根；Δ = 0 时有一个重实根；Δ < 0 时则得到共轭复根。用表格总结这一关系，有助于在考试中快速理清思路。
| Discriminant Δ | Root Type | General Solution Form (for y) |
|---|---|---|
| Δ > 0 | Distinct real r₁, r₂ | C₁ er₁ x + C₂ er₂ x |
| Δ = 0 | Repeated r | (C₁ + C₂ x) er x |
| Δ < 0 | Complex α ± iβ | eα x (C₁ cos βx + C₂ sin βx) |
10. Connection to Physical Systems | 与物理系统的联系
term yields a d²x/dt² + b dx/dt + c x = 0, whose solution type depends on damping ratio, governed by the discriminant.
这些无外力方程模拟了无外力作用的物理场景。例如，简谐振动满足 d²x/dt² + ω² x = 0，其特征根为 r = ±iω，解为 x(t) = C₁ cos ωt + C₂ sin ωt。这描述了永久的正弦运动。加入阻尼项后得到 a d²x/dt² + b dx/dt + c x = 0，其解的类型取决于由判别式决定的阻尼比。
11. Common Mistakes and Tips | 常见错误与提示
One frequent error is writing y = C₁ er x + C₂ er x for a repeated root. Remember that this collapses to a single constant and fails to capture the full solution space. Another mistake is mishandling the derivative when applying initial conditions; always compute y'(x) carefully before substituting. Pay close attention to sign errors when forming the characteristic equation from the differential equation.
一个常见错误是对于重根写出 y = C₁ er x + C₂ er x。请记住，这会合并为单一常数，无法覆盖完整的解空间。另一个错误是在应用初始条件时对导数处理不当；在代入之前一定要仔细计算 y'(x)。在从微分方程建立特征方程时要特别留意符号错误。
12. Summary and Key Takeaways | 总结与要点
To master unforced linear differential equations with constant coefficients, always start by identifying the order and writing the characteristic equation. Determine the discriminant, then classify the roots. Choose the appropriate general-solution template from the three standard cases. Finally, use initial conditions to pin down the arbitrary constants. With practice, this process becomes a routine yet powerful tool for modelling dynamic systems.
要掌握常系数齐次线性微分方程，始终从确定阶数并写出特征方程入手。求出判别式，然后对根进行分类。从三种标准情形中选择合适的通解模板。最后利用初始条件确定任意常数。通过练习，这一过程将成为一个常规且功能强大的工具，用于对动态系统进行建模。
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