📚 Ordinary Differential Equations | 常微分方程
Ordinary differential equations (ODEs) are a cornerstone of calculus, linking rates of change to the functions that describe real-world phenomena. In the IB Mathematics: Analysis and Approaches (AA) and Applications and Interpretation (AI) Higher Level courses, ODEs appear in contexts ranging from population models to mechanics. Mastering ODEs not only strengthens analytical thinking but also prepares you for the problem-solving demands of both examination papers and the internal assessment. This article provides a structured guide to the main techniques and applications required by the IB syllabus.
常微分方程是微积分的基石，它将变化率与描述现实世界现象的函数联系起来。在IB数学分析与方法（AA）和数学应用与解释（AI）高级课程中，常微分方程以人口模型、力学等背景出现。掌握常微分方程不仅能够强化分析思维，还能帮助你应对考试试卷和内部评估中解决问题的要求。本文将对IB大纲要求的主要技巧和应用进行结构化梳理。
1. Introduction to Differential Equations | 微分方程简介
A differential equation is an equation that contains an unknown function and one or more of its derivatives. When the unknown function depends on a single independent variable, it is an ordinary differential equation (ODE). If the function depends on more than one variable and involves partial derivatives, it is a partial differential equation (PDE). In IB Mathematics, we focus exclusively on ODEs. A solution to a differential equation is any function that, when substituted into the equation, reduces it to an identity. The general solution contains arbitrary constants, while a particular solution satisfies a given initial condition.
微分方程是包含未知函数及其一个或多个导数的方程。当未知函数只依赖于一个自变量时，它就是常微分方程（ODE）。如果函数依赖于多个变量并涉及偏导数，则称为偏微分方程（PDE）。在IB数学中，我们专门研究常微分方程。微分方程的解是任何一个代入方程后能使方程恒成立的函数。通解包含任意常数，而特解则满足给定的初始条件。
2. Order and Linearity | 阶与线性
The order of a differential equation is the highest derivative that appears in the equation. For example, dy/dx + y = 0 is first order, while d²y/dx² + 3 dy/dx + 2y = 0 is second order. A differential equation is linear if the dependent variable and its derivatives appear only to the power one and are not multiplied together. The equation dy/dx + x²y = eˣ is linear, but y dy/dx + y = 0 is nonlinear because of the product y dy/dx. An ODE is homogeneous if every term involves the dependent variable or its derivatives; otherwise it is non-homogeneous. Recognising these properties helps choose the right solution method.
微分方程的阶是指方程中出现的最高阶导数。例如，dy/dx + y = 0 是一阶方程，而 d²y/dx² + 3 dy/dx + 2y = 0 是二阶方程。如果因变量及其导数只以一次幂出现且不相乘，则称该微分方程为线性方程。方程 dy/dx + x²y = eˣ 是线性的，但 y dy/dx + y = 0 是非线性的，因为含有乘积 y dy/dx。如果每一项都包含因变量或其导数，则该常微分方程是齐次的；否则就是非齐次的。认清这些性质有助于选择正确的解法。
3. Separable Differential Equations | 可分离变量的微分方程
A first-order ODE is called separable if it can be written in the form dy/dx = f(x) g(y). The variables can be separated so that all y terms are on one side and all x terms on the other: (1/g(y)) dy = f(x) dx. Integrating both sides yields the general solution. This is often the first technique taught because it simply requires algebraic manipulation and integration skills.
若一阶常微分方程可以写成 dy/dx = f(x) g(y) 的形式，则称其为可分离变量的方程。我们可以将变量分离，使所有含 y 的项位于一边，所有含 x 的项位于另一边：(1/g(y)) dy = f(x) dx。等式两边积分便得到通解。这往往是首个被教授的解法，因为它只需要代数操作和积分技巧。
For example, solve dy/dx = 2xy. Separate: ∫ (1/y) dy = ∫ 2x dx. Integration gives ln|y| = x² + C. Exponentiating both sides, |y| = e^{x²+C} = e^{C} e^{x²}. Letting A = ± e^{C}, we obtain the general solution y = A e^{x²}.
例如，求解 dy/dx = 2xy。分离变量：∫ (1/y) dy = ∫ 2x dx。积分得 ln|y| = x² + C。两边取指数，|y| = e^{x²+C} = e^{C} e^{x²}。令 A = ± e^{C}，得到通解 y = A e^{x²}。
Solution: y = A ex²
4. First-Order Linear Differential Equations and Integrating Factors | 一阶线性微分方程与积分因子
A first-order linear ODE can be written in the standard form dy/dx + P(x) y = Q(x). To solve it, we multiply through by an integrating factor μ(x) = e^{∫ P(x) dx}. This factor is chosen so that the left-hand side becomes the derivative of a product: d/dx (μ y) = μ Q(x). Integrating both sides with respect to x gives μ y = ∫ μ Q(x) dx, from which y can be found.
一阶线性常微分方程可以写成标准形式 dy/dx + P(x) y = Q(x)。为了求解，我们乘以积分因子 μ(x) = e^{∫ P(x) dx}。选择该因子是为了使左边成为一个乘积的导数：d/dx (μ y) = μ Q(x)。两边对 x 积分得到 μ y = ∫ μ Q(x) dx，由此可求出 y。
Consider the equation dy/dx + (2/x) y = x. Here P(x) = 2/x, so μ = e^{∫ 2/x dx} = e^{2 ln|x|} = x². Multiply the ODE by x²: x² dy/dx + 2x y = x³. The left side is d/dx (x² y). Integrate: x² y = ∫ x³ dx = ¼ x⁴ + C. Hence y = ¼ x² + C x⁻².
考虑方程 dy/dx + (2/x) y = x。此处 P(x) = 2/x，因此 μ = e^{∫ 2/x dx} = e^{2 ln|x|} = x²。将常微分方程乘以 x²：x² dy/dx + 2x y = x³。左边为 d/dx (x² y)。积分：x² y = ∫ x³ dx = ¼ x⁴ + C。于是 y = ¼ x² + C x⁻²。
General solution: y = ¼ x² + C x⁻²
5. Homogeneous First-Order Equations | 一阶齐次方程
An ODE of the form dy/dx = f(y/x) is called homogeneous (in the sense that the right-hand side depends only on the ratio y/x). To solve it, we use the substitution v = y/x, which implies y = vx and dy/dx = v + x dv/dx. The original equation becomes v + x dv/dx = f(v), which is separable in v and x. After solving for v, we replace v by y/x to obtain the solution in terms of x and y.
形如 dy/dx = f(y/x) 的常微分方程称为齐次方程（指右边只依赖于比值 y/x）。为求解，我们采用代换 v = y/x，则有 y = vx 以及 dy/dx = v + x dv/dx。原方程变为 v + x dv/dx = f(v)，这是关于 v 和 x 的可分离变量方程。解出 v 后，将 v = y/x 代回，即得关于 x 和 y 的解。
For example, solve dy/dx = (y + 2x)/x. Rewrite as dy/dx = y/x + 2. Let v = y/x, then v + x dv/dx = v + 2. Cancel v: x dv/dx = 2. Separate: dv = (2/x) dx. Integrate: v = 2 ln|x| + C. Substitute back: y/x = 2 ln|x| + C, so y = 2x ln|x| + Cx.
例如，求解 dy/dx = (y + 2x)/x。改写为 dy/dx = y/x + 2。令 v = y/x，则 v + x dv/dx = v + 2。消去 v：x dv/dx = 2。分离变量：dv = (2/x) dx。积分：v = 2 ln|x| + C。代回：y/x = 2 ln|x| + C，所以 y = 2x ln|x| + Cx。
6. Numerical Methods: Euler’s Method | 欧拉数值方法
When an ODE cannot be solved analytically, numerical approximations come to the rescue. Euler’s method is the simplest numerical procedure, widely examined in IB AI HL. Starting from an initial point (x₀, y₀) and a step size h, successive points are generated by the formula yₙ₊₁ = yₙ + h f(xₙ, yₙ), where f(x,y) = dy/dx. The method uses the slope at the current point to project the next value linearly. Smaller step sizes improve accuracy but increase computation.
当一个常微分方程无法解析求解时，数值近似就能派上用场。欧拉方法是最简单的数值方法，在IB AI HL考试中经常出现。从初始点 (x₀, y₀) 和步长 h 开始，用公式 yₙ₊₁ = yₙ + h f(xₙ, yₙ) 逐次生成新的点，其中 f(x,y) = dy/dx。该方法利用当前点的斜率线性地推出下一个值。步长越小，精度越高，但计算量也越大。
Suppose dy/dx = x + y with y(0) = 1 and h = 0.1. Then x₀=0, y₀=1. Compute f(0,1)=1, so y₁ = 1 + 0.1×1 = 1.1. Next x₁=0.1, f(0.1,1.1)=1.2, then y₂ = 1.1 + 0.1×1.2 = 1.22. This process continues to build an approximate solution curve.
假设 dy/dx = x + y，y(0) = 1，h = 0.1。那么 x₀=0, y₀=1。计算 f(0,1)=1，于是 y₁ = 1 + 0.1×1 = 1.1。接着 x₁=0.1, f(0.1,1.1)=1.2，则 y₂ = 1.1 + 0.1×1.2 = 1.22。重复该过程便可构造出一条近似解曲线。
7. Slope Fields and Graphical Solutions | 斜率场与图形解
A slope field (or direction field) is a graphical representation of a first-order ODE dy/dx = f(x,y). At a grid of points, short line segments are drawn with slope f(x,y). These segments give a qualitative picture of solution curves without solving the equation analytically. In IB examinations, you may be asked to sketch solution trajectories on a given slope field or to match a differential equation with its slope field. Slope fields also illustrate the idea behind Euler’s method, since the line segments represent the direction of movement from each point.
斜率场（或方向场）是一阶常微分方程 dy/dx = f(x,y) 的图形表示。在网格点上，绘制斜率为 f(x,y) 的短线段。这些线段给出了解曲线的定性图像，而无需解析求解方程。在IB考试中，你可能会被要求在给定的斜率场上绘出解的轨迹，或者将微分方程与其斜率场匹配起来。斜率场也体现了欧拉方法背后的思想，因为每条线段都代表从该点出发的走向。
The beauty of slope fields is that initial conditions can be planted like seeds, and the solution curves grow along the direction segments. This visual approach reinforces the concept that differential equations determine families of curves, and specific solutions are selected by initial values.
斜率场的美妙之处在于，初始条件可以像种子一样种下去，解曲线便会沿着方向线段生长。这种直观方法强化了这样一个概念：微分方程决定了一族曲线，而特解则由初始值筛选出来。
8. Second-Order Linear Homogeneous ODEs with Constant Coefficients | 二阶常系数齐次线性微分方程
In IB AA HL, we extend to second-order equations of the form a d²y/dx² + b dy/dx + c y = 0, where a, b, c are constants. The standard approach is to assume a solution of the form y = e^{r x} and substitute to obtain the auxiliary equation (or characteristic equation) a r² + b r + c = 0. The nature of the roots determines the general solution.
在IB AA HL中，我们将方程拓展为如下形式：a d²y/dx² + b dy/dx + c y = 0，其中 a, b, c 是常数。标准方法是假设解的形式为 y = e^{r x}，代入后得到辅助方程（或特征方程）a r² + b r + c = 0。根的性质决定了通解的形式。
| Roots of auxiliary equation | General solution form |
|---|---|
| Real distinct r₁, r₂ | y = C₁ e^{r₁ x} + C₂ e^{r₂ x} |
| Repeated real r | y = (C₁ + C₂ x) e^{r x} |
| Complex conjugate α ± iβ | y = e^{α x} (C₁ cos βx + C₂ sin βx) |
For example, solve y” – 5y’ + 6y = 0. Auxiliary eq: r² – 5r + 6 = 0 → r = 2, 3. General solution: y = C₁ e²ˣ + C₂ e³ˣ.
例如，求解 y” – 5y’ + 6y = 0。辅助方程：r² – 5r + 6 = 0 → r = 2, 3。通解：y = C₁ e²ˣ + C₂ e³ˣ。
The constants C₁ and C₂ are usually determined by initial conditions on y and y’ at a point. This method is powerful for modelling oscillations, damping, and other physical systems.
常数 C₁ 和 C₂ 通常由某点处 y 和 y’ 的初始条件确定。该方法在振荡、阻尼及其他物理系统建模中非常有效。
9. Applications: Exponential Growth and Decay | 指数增长与衰减
One of the most direct applications of ODEs is the model of exponential growth or decay, described by dy/dt = k y. When k > 0, the quantity y grows without bound; when k < 0, it decays toward zero. The general solution is y = y₀ e^{k t}, where y₀ is the initial value at t = 0. This model describes unlimited population growth, radioactive decay, and continuously compounded interest. The half-life t₁/₂ (for decay) or doubling time (for growth) can be found by setting y = ½ y₀ or y = 2 y₀ respectively, yielding t = (ln 2)/|k|.
常微分方程最直接的应用之一是指数增长或衰减模型，由 dy/dt = k y 描述。当 k > 0 时，量 y 无限增长；当 k < 0 时，它衰减趋近于零。通解为 y = y₀ e^{k t}，其中 y₀ 是 t = 0 时的初始值。该模型可以描述无限制的人口增长、放射性衰变以及连续复利。通过令 y = ½ y₀ 或 y = 2 y₀，可分别求得衰减的半衰期 t₁/₂ 或增长的翻倍时间，结果为 t = (ln 2)/|k|。
Exponential decay: y = y₀ e^{-λ t}, half-life t½ = ln 2 / λ
10. Applications: Newton’s Law of Cooling and Mixing Problems | 牛顿冷却定律与混合问题
Newton’s Law of Cooling states that the rate of change of the temperature T of an object is proportional to the difference between T and the ambient temperature Tₐ: dT/dt = -k (T – Tₐ). This is a first-order linear ODE that can be solved using separation or an integrating factor. The solution takes the form T = Tₐ + (T₀ – Tₐ) e^{-k t}. This model is frequently examined in IB papers, often requiring the determination of k from given data and then predicting temperature at a future time.
牛顿冷却定律指出，物体的温度 T 的变化率与 T 和环境温度 Tₐ 之差成正比：dT/dt = -k (T – Tₐ)。这是一阶线性常微分方程，可以用分离变量法或积分因子法求解。其解的形式为 T = Tₐ + (T₀ – Tₐ) e^{-k t}。该模型在IB试卷中经常出现，通常要求根据给定数据求出 k，进而预测未来某个时刻的温度。
Mixing problems deal with a tank containing a solution where a substance enters and exits. If y(t) is the amount of substance in the tank, the net rate dy/dt = rate in – rate out. The result is often a first-order linear ODE, solvable via an integrating factor.
混合问题则涉及一个装有溶液的容器，其中某种物质以一定速率流入和流出。若 y(t) 表示容器内物质的质量，净变化率 dy/dt = 流入速率 – 流出速率。所得方程通常是一阶线性常微分方程，可通过积分因子求解。
11. Coupled Differential Equations (AI HL) | 耦合微分方程组（AI HL）
In IB AI HL, students encounter systems of coupled differential equations, such as the predator-prey model. For instance, dx/dt = ax – bxy, dy/dt = cxy – dy, where x and y are two interacting populations. These systems cannot usually be solved analytically at this level, so numerical methods are employed. Euler’s method can be extended to coupled equations by applying the stepwise update to each variable simultaneously.
在IB AI HL中，学生会遇到耦合微分方程组，例如捕食者-食饵模型：dx/dt = ax – bxy，dy/dt = cxy – dy，其中 x 和 y 是两个相互作用的种群数量。这类方程组在这一阶段通常无法解析求解，因此需要借助数值方法。欧拉方法可以通过同时更新每个变量推广到耦合方程组。
A typical examination task might ask you to use Euler’s method with given initial populations and step size to generate a few iterations, or to interpret phase portraits showing cyclic behavior. The conceptual leap is understanding that each equation contributes to the direction of change for both variables.
典型的考试任务可能要求你利用给定的初始种群和步长，用欧拉方法进行几次迭代，或者解释显示周期行为的相图。概念上的飞跃在于理解每个方程如何影响两个变量的变化方向。
12. Summary and Exam Tips | 总结与备考技巧
To succeed with ODEs on the IB exam, always begin by identifying the type of equation and its order. For first-order equations, determine whether it is separable, linear, or homogeneous. For second-order linear homogeneous equations, write down the auxiliary equation immediately. In numerical problems, keep careful track of decimal places and ensure step sizes are used consistently. Always check that your solution satisfies
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