📚 First order differential equations and Euler’s method | 一阶微分方程与欧拉方法
Differential equations are the mathematical language of change. From modelling population growth to predicting the motion of a pendulum, first order differential equations allow us to connect a function with its rate of change. In IB Mathematics, you will encounter techniques to solve these equations analytically and numerically, with Euler’s method being the simplest way to approximate solutions when an exact formula is hard to find. This deep dive will strengthen your understanding of slope fields, separable and linear equations, and the step-by-step logic behind Euler’s method — all essential for both Paper 2 and Paper 3.
微分方程是描述变化的数学语言。从种群增长模型到钟摆运动的预测，一阶微分方程将函数与其变化率联系起来。在IB数学中，你将学习解析求解和数值求解的技巧，其中欧拉方法是在难以找到精确解时最简单的近似方法。本文深入探讨斜率场、可分离变量与线性一阶方程，以及欧拉方法的逐步逻辑——这些内容对于Paper 2和Paper 3都至关重要。
1. What is a differential equation? | 什么是微分方程？
nd the original function y(x) that satisfies this relationship. In the IB course, you encounter both ordinary differential equations (ODEs) where y depends only on x, and you learn to interpret them graphically through slope fields.
微分方程是包含未知函数及其导数的方程。一阶微分方程仅含有一阶导数，通常写作 dy/dx = f(x, y)。目标是找到满足这一关系的原函数 y(x)。在IB课程中，你会遇到常微分方程（y 仅依赖于 x），并学习通过斜率场对其进行图形解释。
ion into an initial value problem, which is the starting point for both analytical and numerical methods.
一阶常微分方程的通解含有一个任意常数，而特解则通过施加初始条件（例如 y(x₀) = y₀）得到。这就将方程变成了初值问题，是解析方法和数值方法的起点。
2. Solving by separation of variables | 可分离变量法
grate both sides. Always remember to add the constant of integration, and use the initial condition to find the particular solution if required.
当一阶微分方程可以重新排列，使得所有含 y 的项在一侧、所有含 x 的项在另一侧时，我们使用可分离变量法。我们将 dy/dx = g(x)h(y) 改写为 (1/h(y)) dy = g(x) dx，然后对两边积分。务必记得加上积分常数，并用初始条件求出特解（如果需要的话）。
example, consider dy/dx = 3x²y. Separating gives (1/y) dy = 3x² dx. Integrating yields ln|y| = x³ + C, so y = Aex³, where A = ±eC h an initial condition like y(0) = 5, we find A = 5, giving the particular solution y = 5ex³.
例如，考虑 dy/dx = 3x²y。分离变量得 (1/y) dy = 3x² dx。积分得 ln|y| = x³ + C，因此 y = Aex³，其中 A = ±eC。若给定初始条件 y(0) = 5，可得 A = 5，特解为 y = 5ex³。
3. Recognising first order linear equations | 识别一阶线性微分方程
lying the whole equation by this factor transforms the left-hand side into the derivative of (I(x) y), making integration straightforward.
一阶线性微分方程的标准形式为 dy/dx + P(x)y = Q(x)。与可分离变量方程不同，这类方程不一定能直接分离。我们使用积分因子 I(x) = e^(∫ P(x) dx)。将整个方程乘以这个因子后，左侧变为 (I(x) y) 的导数，积分就变得简单直接。
For instance, to solve dy/dx + 2y = e⁻ˣ, we set P(x)=2, so I(x)=e∫2 dx=e2x. Multiplying through gives e2x dy/dx + 2e2x y = ex. The left is d/dx( e2x y ), so we integrate to get e2x y = ex + C, hence y = e⁻ˣ + Ce⁻²ˣ.
例如，求解 dy/dx + 2y = e⁻ˣ，取 P(x)=2，则 I(x)=e∫2 dx=e2x。两边同乘 I(x) 得 e2x dy/dx + 2e2x y = ex。左边是 d/dx( e2x y )，积分得 e2x y = ex + C，因此 y = e⁻ˣ + Ce⁻²ˣ。
4. Slope fields: visualising differential equations | 斜率场：微分方程的可视化
nts, we draw short line segments with slope equal to f(x, y). These segments form a pattern that suggests the shape of solution curves. By starting at an initial point, we can sketch the approximate path of a particular solution without solving the equation analytically.
斜率场（又称方向场）提供了一阶微分方程 dy/dx = f(x, y) 的图形表示。在网格点上，我们画出斜率等于 f(x, y) 的短线段。这些线段构成的图案揭示了解曲线的形状。从初始点出发，我们可以大致描绘出特解的路径，而不需要解析求解方程。
the equation dy/dx = x + y has slope zero along the line y = -x, which helps in sketching.
在IB考试中，你可能需要为给定方程绘制斜率场，或将斜率场与微分方程匹配。关键线索包括 x=0、y=0 时的行为，以及斜率为零（等斜线）的位置。例如，方程 dy/dx = x + y 沿直线 y = -x 的斜率为零，这有助于草图绘制。
5. Euler’s method: the basic idea | 欧拉方法的基本思想
s method is a numerical approach to approximate the solution of an initial value problem y' o step forward from x₀ using small increments h, and at each step use the current slope to estimate the next y-value. This replaces the actual curve by a series of short straight-line segments, giving a polygonal approximation.
欧拉方法是一种数值方法，用于近似初值问题 y' = f(x, y), y(x₀) = y₀ 的解。核心思想是从 x₀ 开始，以小增量 h 逐步向前推进，每一步使用当前的斜率来估计下一个 y 值。这相当于用一系列短直线段替代实际曲线，得到一条折线近似。
the known initial point, you can generate a table of approximate values for the solution at equally spaced points. The smaller the step size h, the more accurate the approximation tends to be.
递推公式为 xₙ₊₁ = xₙ + h, yₙ₊₁ = yₙ + h · f(xₙ, yₙ)。从已知的初始点出发，你可以生成等间距点处解的近似值表格。步长 h 越小，近似解通常越精确。
6. Deriving Euler’s method step by step | 逐步推导欧拉方法
od follows directly from the definition of the derivative. For a small step h, the slope y'(xₙ) ≈ (y(xₙ₊₁) - y(xₙ))/h. Since the differential equation tells us y' yₙ₊₁ ≈ yₙ + h f(xₙ, yₙ). This linear approximation is simply a tangent line extrapolation.
欧拉方法的推导直接来自导数的定义。对于一个小步长 h，斜率 y'(xₙ) ≈ (y(xₙ₊₁) - y(xₙ))/h。由于微分方程给出 y'(xₙ) = f(xₙ, yₙ)，代入得 (yₙ₊₁ - yₙ)/h ≈ f(xₙ, yₙ)，整理得 yₙ₊₁ ≈ yₙ + h f(xₙ, yₙ)。这种线性近似不过是切线外推。
Graphically, at (xₙ, yₙ) we follow the tangent line for a horizontal distance h to reach the next point. This is why Euler’s method is sometimes called the tangent line method. It is the simplest member of the family of Runge-Kutta methods used in many computational tools.
在图形上，在 (xₙ, yₙ) 点我们沿着切线方向水平移动 h 到达下一个点。这就是为什么欧拉方法有时被称为切线法。它是许多计算工具中使用的龙格-库塔方法家族中最简单的一员。
7. Worked example of Euler’s method | 欧拉方法计算实例
Let’s approximate y(1) for the initial value problem y' 5,1.5) = 0.5+1.5 = 2.0, so y₂ = y₁ + 0.5×2.0 = 1.5 + 1.0 = 2.5 at x₂ = 1.0. So y(1) ≈ 2.5.
我们来近似初值问题 y' = x + y，其中 y(0) = 1，使用步长 h = 0.5，求 y(1)。从 x₀ = 0, y₀ = 1 开始。第一步：斜率 f(0,1) = 0+1 = 1，所以 y₁ = y₀ + 0.5×1 = 1.5，此时 x₁ = 0.5。第二步：斜率 f(0.5,1.5) = 0.5+1.5 = 2.0，所以 y₂ = y₁ + 0.5×2.0 = 1.5+1.0 = 2.5，此时 x₂ = 1.0。因此 y(1) ≈ 2.5。
5), (0.50,1.5625), (0.75,1.953125), (1.0,2.44140625). The true analytic solution is y = 2eˣ 366. The approximations are both underestimates, but the smaller step size is much closer.
如果我们将步长减半至 h = 0.25，得到四步：(0.25,1.25), (0.50,1.5625), (0.75,1.953125), (1.0,2.44140625)。真实解析解是 y = 2eˣ - x - 1，算得 y(1) = 2e - 2 ≈ 3.4366。两个近似值都偏小，但较小的步长更为接近。
8. Euler’s method using a spreadsheet or GDC | 使用电子表格或GDC实现欧拉方法
In IB exams, especially Paper 3, you may be asked to implement Euler’s method using a spreadsheet or your graphing calculator. Setting up columns for n, xₙ, yₙ, f(xₙ, yₙ), and then programming the iteration is a common task. Spreadsheets are excellent because you can easily copy formulas down and vary the step size to observe convergence behaviour.
在IB考试中，特别是Paper 3，你可能需要利用电子表格或图形计算器实现欧拉方法。建立 n, xₙ, yₙ, f(xₙ, yₙ) 的列，然后编写递推程序是一项常见任务。电子表格非常出色，因为你可以轻松地复制公式并改变步长，观察收敛行为。
to compare approximations for different h values and discuss how error decreases as h → 0.
在TI-Nspire或类似图形计算器上，你可以使用电子表格页面或编写简短程序。关键是准确性：确保公式引用正确，并准备好对输出结果进行解释。问题可能要求你比较不同 h 值的近似结果，并讨论当 h → 0 时误差如何减小。
9. Error and limitations of Euler’s method | 欧拉方法的误差与局限性
The error in Euler’s method comes from the fact that we follow the tangent line at the starting point of each step, ignoring the change in slope over the interval. This introduces a local truncation error proportional to h² per step and a global error proportional to h. Thus, halving h roughly halves the global error, but requires twice as many steps.
欧拉方法的误差来源于每一步都遵循起点的切线方向，而忽略了区间内斜率的变化。这引入了每步与 h² 成正比的局部截断误差，以及全球误差与 h 成正比。因此，步长减半大约使全球误差减半，但需要两倍的步数。
Euler’s method is a first-order method, meaning its accuracy is limited. It can be unstable for stiff equations or when h is too large. For practical high-accuracy work, improved methods like the midpoint method or the classical Runge-Kutta (RK4) are preferred, but Euler’s method remains essential for understanding the foundational concepts of numerical integration of ODEs.
欧拉方法是一阶方法，意味着其精度有限。对于刚性方程或 h 过大时，它可能不稳定。在实际需要高精度的工作中，更推荐改进的欧拉方法（中点法）或经典的四阶龙格-库塔法，但欧拉方法对于理解常微分方程数值积分的基础概念仍然不可或缺。
10. Coupled differential equations and extensions | 耦合微分方程与扩展
,y), you simply apply the Euler iteration to both equations simultaneously at each step. This makes it a powerful tool for IB Mathematics: analysis and approaches students who explore modelling in their internal assessment.
欧拉方法可以自然地扩展到一阶微分方程组，例如捕食者-猎物模型或疾病传播模型（SIR模型）中遇到的方程组。对于系统 dx/dt = f(t,x,y), dy/dt = g(t,x,y)，只需在每一步同时对两个方程应用欧拉迭代。这使得它成为IB数学：分析与方法学生在内部评估中探索建模的有力工具。
ing them into a system of first order equations. For instance, the second order equation y'' = F(x, y, y') can be turned into two first order equations by setting v = y', giving y' = v and v' = F(x, y, v). This is a preview of techniques used in applied mathematics and physics.
此外，欧拉方法还可用于通过将高阶常微分方程转化为一阶方程组来近似其解。例如，二阶方程 y'' = F(x, y, y') 可通过设 v = y' 转化为两个一阶方程：y' = v 和 v' = F(x, y, v)。这是应用数学和物理学中使用的技术的预演。
11. Exam tips: what examiners are looking for | 考试技巧：考官在寻找什么
In IB Mathematics papers, when tackling questions on first order differential equations or Euler’s method, focus on clear communication. When solving analytically, show the steps of separation or integrating factor clearly and always state the general solution before substituting the initial condition. For slope fields, carefully annotate axes and mark zero-slope lines.
在IB数学试卷中，当解答一阶微分方程或欧拉方法相关题目时，注重清晰的沟通。解析求解时，应清晰展示分离变量或积分因子的步骤，并始终在代入初始条件前声明通解。对于斜率场，仔细标注坐标轴并标出零斜率线。
When presenting Euler’s method calculations, tabulate your values neatly. Label columns, state the formula you are using, and show the working for at least the first two steps. If asked to discuss accuracy, refer to the effect of step size on global error and illustrate with a numerical comparison if appropriate. Remember that correct notation (dy/dx, yₙ, h) and logical flow can earn method marks even if the final answer is slightly off.
在展示欧拉方法计算时，整洁地制表。为各列添加标签，注明使用的公式，并至少展示前两步的计算过程。如果要求讨论精度，应提及步长对全球误差的影响，并在适当时用数值对比加以说明。请记住，正确的符号（dy/dx, yₙ, h）和逻辑流程即使最终答案略有偏差，也能获得方法分。
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