📚 First-Order Differential Equations & Euler’s Method | 一阶微分方程与欧拉法
Differential equations are mathematical tools that describe how a quantity changes with respect to another variable. In IB Mathematics, first-order differential equations and Euler’s method form a core part of both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses, bridging pure theory with real-world modeling.
微分方程是描述一个量相对于另一个变量如何变化的数学工具。在IB数学中，一阶微分方程与欧拉法是数学分析与方法（AA）和应用与解释（AI）课程的核心内容，将纯理论与现实建模紧密连接。
1. What Is a First-Order Differential Equation? | 什么是一阶微分方程
A first-order differential equation involves only the first derivative dy/dx, and can often be written in the form dy/dx = f(x, y). The order of a differential equation is determined by the highest derivative present.
一阶微分方程只涉及一阶导数 dy/dx，通常可以写成 dy/dx = f(x, y) 的形式。微分方程的阶数取决于方程中出现的最高阶导数。
- General form: F(x, y, dy/dx) = 0
- If dy/dx can be isolated, we write dy/dx = f(x, y)
- 一般形式：F(x, y, dy/dx) = 0
- 如果 dy/dx 可以单独分离出来，可写成 dy/dx = f(x, y)
A solution may be a general solution containing an arbitrary constant, or a particular solution that satisfies an initial condition such as y(x₀) = y₀.
解可以是含有任意常数的通解，也可以是满足初始条件 y(x₀) = y₀ 的特解。
2. Separable Differential Equations | 可分离变量微分方程
If a first-order equation can be written as dy/dx = g(x) · h(y), we say it is separable. We rearrange the equation so that all terms involving y appear with dy, and all terms involving x appear with dx.
如果一阶方程能写成 dy/dx = g(x) · h(y) 的形式，我们称它是可分离变量的。我们将方程重新整理，使所有含 y 的项与 dy 放在一起，所有含 x 的项与 dx 放在一起。
∫ 1/h(y) dy = ∫ g(x) dx + C
This method reduces the differential equation to two separate integrations. Remember to include the constant of integration.
这种方法将微分方程转化为两个独立的积分。别忘了加上积分常数。
3. Worked Example: Separable Equation | 例题：可分离变量方程
Solve the differential equation dy/dx = x²/y, given y(1) = 2.
求解微分方程 dy/dx = x²/y，已知 y(1) = 2。
Separate the variables: y dy = x² dx.
分离变量：y dy = x² dx。
Integrate both sides: y²/2 = x³/3 + C.
两边积分：y²/2 = x³/3 + C。
Apply the initial condition: (2)²/2 = (1)³/3 + C ⇒ 2 = 1/3 + C ⇒ C = 5/3.
代入初始条件：(2)²/2 = (1)³/3 + C ⇒ 2 = 1/3 + C ⇒ C = 5/3。
y²/2 = x³/3 + 5/3 ⇒ y² = 2x³/3 + 10/3
Since y(1) = 2 > 0, we take the positive square root.
因为 y(1) = 2 > 0，我们取正平方根。
4. Linear First-Order Equations and Integrating Factors | 线性一阶方程与积分因子
A linear first-order differential equation has the standard form dy/dx + P(x)y = Q(x). These equations cannot always be solved by separation, so we use an integrating factor.
线性一阶微分方程的标准形式为 dy/dx + P(x)y = Q(x)。这类方程不一定能通过分离变量求解，因此我们使用积分因子。
Integrating factor: I(x) = e^(∫ P(x) dx)
Multiplying both sides of the equation by I(x) converts the left-hand side into the derivative of I(x)y.
将方程两边乘以 I(x)，可将左边转化为 I(x)y 的导数。
d/dx [I(x)y] = I(x)Q(x)
Then integrate both sides and solve for y.
然后两边积分并解出 y。
5. Worked Example: Integrating Factor | 例题：积分因子
Solve dy/dx + 2xy = x, with y(0) = 1.
求解 dy/dx + 2xy = x，已知 y(0) = 1。
Here P(x) = 2x, so I(x) = e^(∫ 2x dx) = e^(x²).
这里 P(x) = 2x，所以 I(x) = e^(∫ 2x dx) = e^(x²)。
Multiplying: e^(x²) dy/dx + 2x e^(x²) y = x e^(x²).
两边乘以积分因子：e^(x²) dy/dx + 2x e^(x²) y = x e^(x²)。
The left side becomes d/dx [e^(x²) y]. Integrate both sides: e^(x²) y = ∫ x e^(x²) dx.
左边变为 d/dx [e^(x²) y]。两边积分：e^(x²) y = ∫ x e^(x²) dx。
Use substitution u = x², du = 2x dx ⇒ ∫ x e^(x²) dx = e^(x²)/2 + C.
使用换元 u = x²，du = 2x dx ⇒ ∫ x e^(x²) dx = e^(x²)/2 + C。
Thus y = 1/2 + C e^(−x²). With y(0) = 1, we get 1 = 1/2 + C ⇒ C = 1/2.
因此 y = 1/2 + C e^(−x²)。由 y(0) = 1 得 1 = 1/2 + C ⇒ C = 1/2。
y = 1/2 + 1/2 e^(−x²)
6. Euler’s Method: Concept | 欧拉法：概念
Euler’s method is a numerical technique used to approximate the solution of an initial value problem dy/dx = f(x, y), y(x₀) = y₀. It is especially useful when an analytic solution is difficult or impossible to find.
欧拉法是一种数值方法，用于近似求解初值问题 dy/dx = f(x, y)，y(x₀) = y₀。当解析解难以求出或根本不存在时，它尤其有用。
The method uses the tangent line at a known point to estimate the next point. Given a step size h, the recurrence formulas are:
该方法利用已知点处的切线来估计下一个点。给定步长 h，递推公式为：
xₙ₊₁ = xₙ + h, yₙ₊₁ = yₙ + h · f(xₙ, yₙ)
Each step follows the slope at the current point. Smaller step sizes generally give more accurate approximations but require more computation.
每一步都沿着当前点的斜率前进。步长越小，近似通常越准确，但计算量也越大。
7. Worked Example: Euler’s Method | 例题：欧拉法
Use Euler’s method with step size h = 0.5 to approximate y(1) for the initial value problem dy/dx = x + y, y(0) = 1.
使用欧拉法，取步长 h = 0.5，近似求解初值问题 dy/dx = x + y，y(0) = 1 在 x = 1 处的 y 值。
Step 1: x₀ = 0, y₀ = 1. Slope = f(0, 1) = 0 + 1 = 1.
第1步：x₀ = 0，y₀ = 1。斜率 = f(0, 1) = 0 + 1 = 1。
y₁ = y₀ + h · f(x₀, y₀) = 1 + 0.5 × 1 = 1.5; x₁ = 0.5.
y₁ = y₀ + h · f(x₀, y₀) = 1 + 0.5 × 1 = 1.5；x₁ = 0.5。
Step 2: x₁ = 0.5, y₁ = 1.5. Slope = f(0.5, 1.5) = 0.5 + 1.5 = 2.
第2步：x₁ = 0.5，y₁ = 1.5。斜率 = f(0.5, 1.5) = 0.5 + 1.5 = 2。
y₂ = 1.5 + 0.5 × 2 = 2.5; x₂ = 1.0.
y₂ = 1.5 + 0.5 × 2 = 2.5；x₂ = 1.0。
Therefore the Euler approximation is y(1) ≈ 2.5.
因此欧拉近似为 y(1) ≈ 2.5。
8. Error and Step Size | 误差与步长
Euler’s method introduces two types of error: local truncation error from each linear approximation, and accumulated global error over many steps. The local error is proportional to h², while the global error is proportional to h.
欧拉法会产生两类误差：每一步线性近似产生的局部截断误差，以及多步累积的全局误差。局部误差与 h² 成正比，全局误差与 h 成正比。
| Step size h | Number of steps | Expected accuracy |
| Large | Few | Lower |
| Small | Many | Higher |
In IB exams, you may be asked to perform a fixed number of steps. Always write down each iteration clearly to avoid arithmetic errors.
在IB考试中，你可能会被要求执行固定步数的计算。务必清楚地写出每一步迭代，以避免算术错误。
9. When to Use Each Method | 何时使用每种方法
Separable equations are best when f(x, y) can be factored into a product g(x)h(y). Linear equations with P(x) and Q(x) require an integrating factor. Euler’s method is used when an explicit solution is not required or is impossible to obtain.
当 f(x, y) 可以分解为 g(x)h(y) 的乘积时，分离变量法最合适。具有 P(x) 和 Q(x) 的线性方程需要使用积分因子。当不需要显式解或无法获得显式解时，使用欧拉法。
- dy/dx = x²/y → separable
- dy/dx + 2xy = x → linear, integrating factor
- dy/dx = sin(xy) → no simple analytic method, use Euler
- dy/dx = x²/y → 可分离变量
- dy/dx + 2xy = x → 线性，积分因子
- dy/dx = sin(xy) → 无简单解析方法，使用欧拉法
10. Common IB Exam Tips | IB考试常见提示
Always check whether the equation is separable before attempting an integrating factor. For Euler’s method, keep a table with columns for xₙ, yₙ, f(xₙ, yₙ), and the next y value.
在尝试积分因子之前，先检查方程是否可分离变量。对于欧拉法，用表格记录 xₙ、yₙ、f(xₙ, yₙ) 以及下一个 y 值。
When integrating, do not forget the constant C. When applying an initial condition, solve for C exactly. When using Euler’s method, use the slope at the beginning of each interval, not the midpoint.
积分时不要忘记常数 C。代入初始条件时，精确求解 C。使用欧拉法时，使用每个区间起点的斜率，而不是中点。
11. Real-World Applications | 实际应用
First-order differential equations model population growth, radioactive decay, Newton’s law of cooling, and financial investments. Euler’s method allows computers to simulate these systems even when exact formulas are unknown.
一阶微分方程可以模拟人口增长、放射性衰变、牛顿冷却定律和金融投资。即使没有精确公式，欧拉法也能让计算机模拟这些系统。
For example, the logistic equation dy/dt = r y (1 − y/K) is separable and models limited population growth. In finance, the accumulation of continuously compounded interest follows dy/dt = r y.
例如，逻辑斯谛方程 dy/dt = r y (1 − y/K) 是可分离的，用于模拟有限的人口增长。在金融领域，连续复利的累积遵循 dy/dt = r y。
12. Summary | 总结
First-order differential equations can be solved analytically by separating variables or using an integrating factor. When analytic methods fail, Euler’s method provides a simple numerical approximation. Understanding both approaches prepares you for a wide range of IB exam questions.
一阶微分方程可以通过分离变量或使用积分因子来解析求解。当解析方法无效时，欧拉法提供了简单的数值近似。掌握这两种方法，可以帮助你应对广泛的IB考试题型。
Practice by rewriting each equation into its standard form first, then choosing the appropriate method. With consistent practice, differential equations will become one of the most rewarding topics in IB Mathematics.
练习时先将每个方程改写为标准形式，然后选择合适的方法。通过持续练习，微分方程将成为IB数学中最有收获的专题之一。
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