Differentials and the Solution of First-Order Equations | 微分与一阶微分方程解法

📚 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。

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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 ‘(x) dx. Here dx represents an infinitesimally small change in the independent variable x, and dy gives the corresponding change along the tangent line at x. In geometric terms, dy measures the vertical rise of the tangent when 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。

We also write the differential of the independent variable as dx = Δx. This notation allows us to propagate differentials 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。

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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) – f(a)]/h exists. Differentiability implies continuity, but continuity alone does not guarantee differentiability – corners, cusps, and vertical tangents are classic counterexamples. The practical power of the differential lies in linear approximation: for small Δx, Δy ≈ dy. That is, f(a + Δx) ≈ f(a) + f ‘(a) Δx. This is the tangent-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。

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3. Differentials and Error Propagation | 微分与误差传播

When a quantity is calculated from measured inputs that carry small uncertainties, differentials provide a clean estimate of the resulting error. Suppose a quantity Q depends on a measured variable x, and the measurement has an error dx. Then the absolute error in Q is approximately |dQ| = |Q ‘(x) dx|. For products and powers, relative error analysis becomes especially elegant. If A = π r² and the radius has an error dr, then dA = 2π r dr, and the relative error dA/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

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4. First-Order Differential Equations: An Introduction | 一阶微分方程简介

A first-order ordinary differential equation (ODE) has the general form dy/dx = f(x, y), where f is a given function of two variables. A solution is any function y = φ(x) that satisfies the equation on some interval. The family of all solutions is called the general solution and contains an arbitrary 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。

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5. Separable Equations | 可分离变量方程

A differential equation is called separable if it can be written as dy/dx = g(x) h(y) or, equivalently, (1/h(y)) dy = g(x) dx, provided h(y) ≠ 0. The solution is then obtained by integrating both sides: ∫ (1/h(y)) dy = ∫ g(x) dx + 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。

Example: Solve dy/dx = 3x² y. Separating gives (1/y) dy = 3x² dx. Integrating both sides yields ln|y| = x³ + C, hence |y| = e^(x³ + C) = e^C e^(x³). Letting K = ±e^C gives the general solution y = K e^(x³).

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| = e^(x³ + C) = e^C e^(x³)。令 K = ±e^C 即得通解 y = K e^(x³)。

Always check for missing constant solutions: the step dividing by h(y) discards solutions 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。

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6. First-Order Linear Equations and Integrating Factors | 一阶线性方程与积分因子

A first-order linear ODE has the standard form dy/dx + P(x) y = Q(x), where P and Q are functions of x alone. The key to solving it is the integrating factor μ(x) = e^(∫ P(x) dx). Multiplying the entire equation by μ transforms the left-hand side into the derivative of the product μ y: d/dx (μ y) = μ Q(x). Integrating 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? We require dμ/dx = μ P(x), which is a separable equation for μ with 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) = e^(2 ln x) = x². Multiply: x² dy/dx + 2x y = x⁵. The 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) = e^(2 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。

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7. Homogeneous First-Order Equations | 齐次一阶方程

An equation is called homogeneous if it can be written in the form dy/dx = F(y/x), where the right-hand side depends only on the ratio y/x. The standard substitution v = y/x, i.e. y = v x, transforms the equation into a separable one for v. Since 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。

Example: dy/dx = (x² + y²)/(x y). Divide numerator and denominator by x²: dy/dx = (1 + (y/x)²)/(y/x). Let v = y/x, then v + x dv/dx = (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。

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8. Bernoulli Equations | 伯努利方程

A Bernoulli equation is a nonlinear first-order ODE of the form dy/dx + P(x) y = Q(x) yⁿ, where n is a real constant, n ≠ 0,1. Although nonlinear, it can be reduced to a linear equation by the substitution z = y^(1–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 = y^(1–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–n) Q(x), which is linear 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 = y^(1–2) = 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) = e^x, so e^x z = ∫ –x e^x dx. Integration by parts gives –x e^x + e^x + 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 = y^(1–2) = 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) = e^x，所以 e^x z = ∫ –x e^x dx。分部积分得 –x e^x + e^x + C，因此 z = –x + 1 + C e^(–x)。最终 y = 1/z = 1/(1 – x + C e^(–x))。

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9. Slope Fields and Qualitative Analysis | 斜率场与定性分析

Even when a first-order ODE cannot be solved analytically, its behaviour can be studied through slope fields. Given dy/dx = f(x,y), we evaluate f at a grid of points and draw short line segments with the corresponding slopes. This graphical representation reveals the general shape of solution curves 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。

For instance, the equation dy/dx = x + y yields 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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