📚 Differentials and the Solution of First-Order Equations | 微分与一阶微分方程解法
Differential calculus equips us with the notion of the differential, a precise way to describe how small changes in input relate to changes in output. When we combine this concept with first-order differential equations, we gain the power to model dynamic systems – population growth, radioactive decay, cooling processes, and countless other natural phenomena. This article provides a thorough walkthrough of the definitions, techniques, and applications required for IB Mathematics, from basic differentials to the analytical solution of first-order equations.
微分学为我们提供了微分的概念，这是一种精确描述输入微小变化如何影响输出变化的方法。当我们将这一概念与一阶微分方程相结合，就获得了对动态系统建模的能力——人口增长、放射性衰变、冷却过程以及无数其他自然现象。本文全面梳理IB数学所需的定义、解题技巧与应用，从基本微分到一阶方程的解析求解。
1. What is a Differential? | 什么是微分？
If y = f(x) is a differentiable function, the differential dy is defined as dy = f ' en x moves by dx, while the actual change Δy is the rise on the curve. The equality dy = f 'x) dx formalises the idea that the derivative can be treated as a ratio of infinitesimals.
若 y = f(x) 是可微函数，则微分 dy 定义为 dy = f '(x) dx。这里 dx 表示自变量 x 的无穷小变化，dy 给出在 x 处切线方向上的相应变化。从几何角度看，dy 衡量的是当 x 移动 dx 时切线的纵向增量，而真实变化 Δy 是曲线上的增量。等式 dy = f '(x) dx 使得导数可以被当作无穷小之比的这一想法形式化。
ifferentials through more complicated relationships: if u = g(x) and v = h(u), then dv = h '(u) du = h '(g(x)) g '(x) dx, which is simply the chain rule in differential form.
我们也把自变量的微分记作 dx = Δx。这种记法使我们可以将微分在更复杂的函数关系中传递：若 u = g(x) 且 v = h(u)，则 dv = h '(u) du = h '(g(x)) g '(x) dx，这正是微分形式的链式法则。
2. Differentiability and Linear Approximation | 可微性与线性近似
A function f is differentiable at a point a if the limit f '(a) = lim(h→0) [f(a+h) – rentiability implies continuity, but continuity alone does not guarantee differentiability – rential lies in linear approximation: for small Δx, Δy ≈ dy. That is, f(a + Δx) ≈ f(a) + f ' -line approximation, and it forms the basis for error propagation and numerical estimates.
函数 f 在点 a 处可微，当且仅当极限 f '(a) = lim(h→0) [f(a+h) – f(a)]/h 存在。可微性蕴含连续性，但连续性本身不能保证可微——尖点、角点和垂直切线都是典型反例。微分的实际威力在于线性近似：当 Δx 很小时，Δy ≈ dy。即 f(a + Δx) ≈ f(a) + f '(a) Δx。这就是切线近似，它构成了误差传递与数值估计的基础。
Example: Estimate √4.1 using the differential. Let f(x) = √x, with a = 4, Δx = 0.1. Then f '(x) = 1/(2√x), f '(4) = 1/4. The approximation gives √4.1 ≈ 2 + (1/4)×0.1 = 2.025. The true value is about 2.0248, showing excellent agreement.
例题：用微分估算 √4.1。设 f(x) = √x，取 a = 4，Δx = 0.1。则 f '(x) = 1/(2√x)，f '(4) = 1/4。近似给出 √4.1 ≈ 2 + (1/4)×0.1 = 2.025。真实值约为 2.0248，吻合极好。
3. Differentials and Error Propagation | 微分与误差传播
able x, and the measurement has an error dx. Then the absolute error in Q is approximately |dQ| = |Q '(x) dx| /A = 2 dr/r, meaning the percentage error in area is twice the percentage error in radius.
当某个量是由带有微小不确定度的测量值计算而来时，微分可以干净地估计结果误差。假设量 Q 依赖于被测量 x，且该测量具有误差 dx，则 Q 的绝对误差近似为 |dQ| = |Q '(x) dx|。对于乘积与幂函数，相对误差分析变得格外简洁。若 A = π r² 且半径有误差 dr，则 dA = 2π r dr，相对误差 dA/A = 2 dr/r，意味着面积的百分误差是半径百分误差的两倍。
| Error type | Formula | 误差类型 | 公式 |
| Absolute error | ΔQ ≈ dQ = Q '(x) dx | 绝对误差 | ΔQ ≈ dQ = Q '(x) dx |
| Relative error | dQ / Q | 相对误差 | dQ / Q |
4. First-Order Differential Equations: An Introduction | 一阶微分方程简介
ry constant C. If we additionally specify an initial condition y(x₀) = y₀, the resulting problem is an initial value problem (IVP), and its solution is a particular solution. Geometrically, the differential equation assigns a slope to every point (x,y), constructing a slope field.
一阶常微分方程的一般形式为 dy/dx = f(x, y)，其中 f 是给定的二元函数。解是指任何在某个区间上满足该方程的函数 y = φ(x)。所有解构成的族称为通解，含有一个任意常数 C。若额外指定一个初始条件 y(x₀) = y₀，得到的就是初值问题，其解为特解。从几何上看，微分方程为平面上的每一点 (x,y) 赋予一个斜率，从而构造出斜率场。
Our goal is to develop analytical methods for several common families of first-order ODEs: separable, linear, homogeneous, and Bernoulli equations. Each method transforms the original equation into a form that can be integrated directly.
我们的目标是发展几类常见一阶常微分方程的解析解法：可分离变量方程、线性方程、齐次方程和伯努利方程。每种方法都将原方程转化为可直接积分求解的形式。
5. Separable Equations | 可分离变量方程
+ C. After integration, we solve for y explicitly if possible, or leave the solution in implicit form.
若一个微分方程可以写成 dy/dx = g(x) h(y)，或等价地写成 (1/h(y)) dy = g(x) dx（只要 h(y) ≠ 0），则称之为可分离变量方程。然后通过两边积分求解：∫ (1/h(y)) dy = ∫ g(x) dx + C。积分后，若可能则解出 y 的显式表达式，或保留隐式形式。
Solve dy/dx = 3x² y. Separating gives (1/y) dy = 3x² dx. Integrating both sides yields ln|y| = x³ + C, hence |y| = ex³ + C = eC ex³. Letting K = ±eC gives the general solution y = K ex³.
例题：求解 dy/dx = 3x² y。分离变量得 (1/y) dy = 3x² dx。两边积分得 ln|y| = x³ + C，因此 |y| = ex³ + C = eC ex³。令 K = ±eC 即得通解 y = K ex³。
ions where h(y) = 0. In the example, y = 0 is also a solution, which corresponds to K = 0.
务必检查遗失的常值解：除以 h(y) 的步骤会漏掉使 h(y) = 0 的解。上述例题中，y = 0 也是解，对应 K = 0。
6. First-Order Linear Equations and Integrating Factors | 一阶线性方程与积分因子
ntegrating both sides yields μ y = ∫ μ Q(x) dx + C, and finally y = (1/μ)[∫ μ Q(x) dx + C].
一阶线性常微分方程的标准形式为 dy/dx + P(x) y = Q(x)，其中 P 和 Q 都只依赖于 x。求解的关键是积分因子 μ(x) = e^(∫ P(x) dx)。用 μ 乘以整个方程后，左侧变为乘积 μ y 的导数：d/dx (μ y) = μ Q(x)。两边积分得 μ y = ∫ μ Q(x) dx + C，最后解出 y = (1/μ)[∫ μ Q(x) dx + C]。
Why does this work? ith solution μ = e^(∫ P(x) dx). This choice guarantees the product rule simplifies neatly.
为什么可以这样做？我们需要 dμ/dx = μ P(x)，这是关于 μ 的可分离变量方程，解为 μ = e^(∫ P(x) dx)。这一选择确保了乘法法则的简洁化简。
Example: Solve dy/dx + (2/x) y = x³ for x > 0. Here P(x) = 2/x, so μ = e∫ 2/x dx = e2 ln x he left side is d/dx (x² y). Integrate: x² y = ∫ x⁵ dx = x⁶/6 + C, giving y = x⁴/6 + C/x².
例题：求解 dy/dx + (2/x) y = x³（x > 0）。这里 P(x) = 2/x，因此 μ = e∫ 2/x dx = e2 ln x = x²。相乘：x² dy/dx + 2x y = x⁵。左侧即 d/dx (x² y)。积分：x² y = ∫ x⁵ dx = x⁶/6 + C，得 y = x⁴/6 + C/x²。
7. Homogeneous First-Order Equations | 齐次一阶方程
e dy/dx = v + x dv/dx, the original equation becomes v + x dv/dx = F(v), or x dv/dx = F(v) – v, which is separable.
如果一个方程可以写成 dy/dx = F(y/x) 的形式，即右边仅依赖于比值 y/x，则称之为齐次方程。标准代换 v = y/x，即 y = v x，可将该方程转化为关于 v 的可分离变量方程。由 dy/dx = v + x dv/dx，原方程化为 v + x dv/dx = F(v)，即 x dv/dx = F(v) – v，这是一个可分离变量方程。
(1 + v²)/v. Rearranging: x dv/dx = 1/v. Separating: v dv = (1/x) dx, integrate: v²/2 = ln|x| + C. Substituting back: (y/x)²/2 = ln|x| + C, or y² = 2x² ln|x| + K x².
例题：dy/dx = (x² + y²)/(x y)。分子分母同除以 x²：dy/dx = (1 + (y/x)²)/(y/x)。令 v = y/x，则 v + x dv/dx = (1 + v²)/v。整理得 x dv/dx = 1/v。分离变量：v dv = (1/x) dx，积分：v²/2 = ln|x| + C。代回： (y/x)²/2 = ln|x| + C，或 y² = 2x² ln|x| + K x²。
8. Bernoulli Equations | 伯努利方程
0,1. Although nonlinear, it can be reduced to a linear equation by the substitution z = y1–n. Differentiating gives dz/dx = (1–n) y–n dy/dx. Multiplying the original equation by (1–n) y–n yields a linear equation in z.
伯努利方程是形如 dy/dx + P(x) y = Q(x) yⁿ 的非线性一阶常微分方程，其中 n 为实常数，且 n ≠ 0,1。该方程虽为非线性，但可通过代换 z = y1–n 化为线性方程。求导得 dz/dx = (1–n) y–n dy/dx。将原方程乘以 (1–n) y–n 即得关于 z 的线性方程。
After substitution, the equation becomes dz/dx + (1–n) P(x) z = (1– r in z and can be solved using an integrating factor. Once z is found, recover y = z^(1/(1–n)).
代换后，方程化为 dz/dx + (1–n) P(x) z = (1–n) Q(x)，这是关于 z 的线性方程，可用积分因子法求解。求得 z 后，通过 y = z^(1/(1–n)) 还原 y。
Example: dy/dx – y = x y². Here P(x) = –1, Q(x) = x, n = 2. Let z = y1–2 = y–1. Then dz/dx = –y–2 dy/dx. Multiplying the original equation by –y–2 gives –y–2 dy/dx + y–1 = –x. That is dz/dx + z = –x. Solve this linear equation: μ = e∫1 dx = ex, so ex z = ∫ –x ex dx. Integration by parts gives –x ex + ex + C, hence z = –x + 1 + C e–x. Finally y = 1/z = 1/(1 – x + C e–x).
例题：dy/dx – y = x y²。此处 P(x) = –1，Q(x) = x，n = 2。令 z = y1–2 = y–1。则 dz/dx = –y–2 dy/dx。原方程乘以 –y–2 得 –y–2 dy/dx + y–1 = –x，即 dz/dx + z = –x。解该线性方程：μ = e∫1 dx = ex，所以 ex z = ∫ –x ex dx。分部积分得 –x ex + ex + C，因此 z = –x + 1 + C e–x。最终 y = 1/z = 1/(1 – x + C e–x)。
9. Slope Fields and Qualitative Analysis | 斜率场与定性分析
urves and helps identify equilibrium solutions where dy/dx = 0. It also allows us to sketch approximate solutions passing through a given initial point without solving the equation explicitly.
即便一阶常微分方程无法解析求解，也可以通过斜率场研究其行为。给定 dy/dx = f(x,y)，我们在网格点处计算 f 的值，并画出相应斜率的短线段。这种图形表示揭示了积分曲线的大致形状，并有助于识别使 dy/dx = 0 的平衡解。它也使我们无需显式求解方程，便可绘制通过给定初始点的近似解曲线。
elds a slope field in which solution curves trend upward as x and y increase. The line y = –x – 1 makes dy/dx = –1, while along y = –x the slope is zero. By tracing the field, we imagine how particular solutions evolve.
例如，方程 dy/dx = x + y 产生的斜率场中，当 x 和 y 增大时，解曲线上扬。直线 y = –x – 1 上斜率为 –1，而沿着 y = –x 斜率为零。通过追踪斜率场，我们可以想象特解的演化
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