Differential Equations Exam Essentials | 微分方程考点精讲

📚 Differential Equations Exam Essentials | 微分方程考点精讲

Differential equations form a core part of advanced mathematics, linking rates of change to the functions that model real-world phenomena. In both IB (Analysis & Approaches HL) and WJEC A-level Mathematics, you are expected to solve first-order ODEs analytically, interpret slope fields, apply Euler’s method, and use differential equations to model growth, decay, and mixing problems. This guide systematically covers every key concept, technique, and exam-ready strategy you need to master.

微分方程是高等数学的核心内容，它将变化率与描述现实世界现象的函数联系起来。在 IB（分析与方法 HL）和 WJEC A-level 数学中，你需要掌握一阶常微分方程的解析求解、斜率场的解读、欧拉方法的应用，并能用微分方程建立增长、衰减及混合模型。本文将系统地梳理每个关键概念、解题技巧和应考策略，助你全面攻克这一考点。

1. What Is a Differential Equation? | 什么是微分方程？

A differential equation (DE) is an equation involving an unknown function and one or more of its derivatives. The order of a DE is the highest derivative present, and the degree is the power to which that highest derivative is raised after rationalisation.

微分方程是包含未知函数及其导数（或微分）的方程。方程的阶是指出现的最高阶导数的阶数，次是指在有理化后最高阶导数的幂次。

In both IB and WJEC, you will primarily work with first-order ordinary differential equations (dy/dx appears, but no partial derivatives). The general form is dy/dx = f(x, y) or M(x,y)dx + N(x,y)dy = 0.

在 IB 和 WJEC 课程中，你主要处理一阶常微分方程（仅出现 dy/dx，无偏导数）。其一般形式为 dy/dx = f(x, y) 或 M(x,y)dx + N(x,y)dy = 0。

2. Separable Differential Equations | 可分离变量微分方程

If a first-order DE can be written in the form g(y) dy = h(x) dx, it is separable. Integration on both sides then yields the general solution. Remember to include the constant of integration +C on one side only.

当一阶微分方程可以写成 g(y) dy = h(x) dx 的形式时，它就是可分离变量的。对两边同时积分即可得到通解。注意只需在一边加上积分常数 +C。

Examiners often test separation by requiring algebraic manipulation first. For instance, dy/dx = xy + x can be factored as dy/dx = x(y+1), then separated: dy/(y+1) = x dx. Always check for constant solutions like y = −1 that might be lost during separation; these are often singular solutions.

考官常会通过要求先进行代数变形来测试分离方法。例如 dy/dx = xy + x，可以先因式分解为 dy/dx = x(y+1)，再分离变量：dy/(y+1) = x dx。分离变量时，要检查是否会丢失常数解，比如 y = −1，这类解往往是奇异解。

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

A first-order linear DE has the standard form dy/dx + P(x)y = Q(x). The general solution is found by multiplying through by the integrating factor (IF): μ(x) = e∫P(x)dx. Then the left-hand side becomes the derivative of μ(x)y. Integrate both sides to obtain y.

一阶线性微分方程的标准形式为 dy/dx + P(x)y = Q(x)。求通解的步骤是先乘以积分因子 μ(x) = e∫P(x)dx，此时左边恰好是 μ(x)y 的导数。对两边积分即可解出 y。

In IB exams, you are given complicated P(x) and Q(x), requiring accurate integration and use of initial conditions. WJEC papers commonly include context-based linear DEs, like mixing or electrical circuits. After integrating, apply the boundary condition to find the particular solution.

在 IB 考试中，有时会给出较复杂的 P(x) 和 Q(x)，需要精确积分并代入初始条件。WJEC 试卷中常见情境类线性方程，如混合问题或电路问题。积分后，应用边界条件来求出特解。

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

A DE of the type dy/dx = f(y/x) is called homogeneous. Use the substitution y = vx (where v is a function of x), so dy/dx = v + x dv/dx. The equation reduces to a separable one in terms of v and x.

形如 dy/dx = f(y/x) 的方程称为齐次方程。作代换 y = vx（v 是 x 的函数），则 dy/dx = v + x dv/dx。方程便化为关于 v 和 x 的可分离变量方程。

After solving for v, replace v with y/x to return to the original variables. IB HL sometimes asks students to recognise homogeneous equations without prompting. Remember to simplify f(y/x) carefully before separating, and avoid algebraic slips when back-substituting.

解出 v 后，将 v = y/x 代回，还原为原变量。IB HL 有时要求学生能自行识别齐次方程。在分离变量前要仔细化简 f(y/x)，并在回代时避免代数错误。

5. Exact Differential Equations and Partial Derivatives | 恰当方程与偏导数（扩展考点）

Some IB HL courses or WJEC Further Mathematics modules introduce exact equations: M(x,y)dx + N(x,y)dy = 0 is exact if ∂M/∂y = ∂N/∂x. The solution is a potential function F(x,y) = C, where Fx = M and Fy = N. This topic deepens understanding but is less common in standard papers; however, it can be a helpful unifying perspective.

部分 IB HL 课程或 WJEC 进阶数学模块会引入恰当方程：若 ∂M/∂y = ∂N/∂x，则 M(x,y)dx + N(x,y)dy = 0 为恰当方程。其解为势函数 F(x,y) = C，且满足 Fx = M，Fy = N。该知识点虽不常直接考查，但有助于建立微分方程的统一视角。

6. Initial Conditions and Particular Solutions | 初始条件与特解

After finding the general solution (with an arbitrary constant), an initial condition y(x₀) = y₀ determines the unique particular solution. In applications, this represents a known state at a given time.

求得含有任意常数的通解后，利用初始条件 y(x₀) = y₀ 可确定唯一的特解。在实际问题中，初始条件代表着某个特定时刻的已知状态。

Always present the final answer in the simplest explicit form if requested, e.g. y = 3e⁻²ˣ + 1 rather than leaving it implicit. IB exam questions often ask for the domain of the solution as well, based on the context or denominator restrictions.

如题目要求，尽可能将最终答案化简为显式形式，例如 y = 3e⁻²ˣ + 1，而非保留隐式形式。IB 考试中，有时还会要求根据情境或分母限制给出解的定义域。

7. Slope Fields and Solution Curves | 斜率场与解曲线

A slope field (direction field) is a graphical representation of dy/dx = f(x,y). Short line segments are drawn at grid points with slope f(x,y). Solution curves are drawn to follow these segments, providing a visual family of solutions.

斜率场（方向场）是 dy/dx = f(x,y) 的图形表示。在网格点处画出斜率为 f(x,y) 的短线段。解曲线就是跟随这些线段的走向绘制的，从而可视化一族解。

Both IB and WJEC require you to sketch slope fields for given differential equations, usually over a small grid like −2 ≤ x,y ≤ 2. You must also sketch particular solution curves through a given point. In digital exams, you might be asked to match slope fields with DEs.

IB 和 WJEC 均要求你为给定的微分方程绘制斜率场，通常在 −2 ≤ x,y ≤ 2 的小网格内。还需要画出经过给定点的特解曲线。在数字化考试中，可能出现匹配斜率场与微分方程的题目。

8. Euler’s Method for Numerical Approximation | 欧拉数值近似法

When a DE cannot be solved analytically, Euler’s method provides an approximate solution. Starting at (x₀, y₀) with step size h, the next point is given by yₙ₊₁ = yₙ + h·f(xₙ, yₙ). Repeating the process builds a polygonal approximation to the true solution curve.

当微分方程无法解析求解时，欧拉方法可以给出数值近似解。从初始点 (x₀, y₀) 出发，步长为 h，下一个点的纵坐标为 yₙ₊₁ = yₙ + h·f(xₙ, yₙ)。重复此步骤可构造出真实解曲线的折线近似。

IB HL exams present Euler’s method in tables, requiring 2-3 iterations. You must be fluent in calculator use for the repetitive calculations and understand that smaller step size generally improves accuracy but increases computational effort. WJEC may include a question on the accuracy of approximation.

IB HL 考卷常以表格形式呈现欧拉方法，需进行 2−3 步迭代。应熟练使用计算器进行重复运算，并理解较小的步长通常能提高精度但会增加计算量。WJEC 可能会出现关于近似精度评判的题目。

9. Modelling Growth and Decay | 增长与衰减模型

The natural growth/decay model dy/dt = ky leads to y = Ceᵏᵗ. If k > 0, it’s exponential growth; if k < 0, exponential decay. Given two data points, you can solve for k and C. The half-life for decay is tₕ = (ln 2)/|k|; doubling time for growth is t_d = (ln 2)/k.

自然增长/衰减模型 dy/dt = ky 的解为 y = Ceᵏᵗ。若 k > 0，为指数增长；若 k < 0，为指数衰减。给定两个数据点即可解出 k 和 C。衰减的半衰期为 tₕ = (ln 2)/|k|；增长的倍增时间为 t_d = (ln 2)/k。

Common exam problems include population growth, radioactive decay, and continuously compounded interest. You may need to convert between half-life and rate constant k. Always define variables clearly and check units.

常见考题包括人口增长、放射性衰变和连续复利。可能需要在半衰期和速率常数 k 之间进行转换。务必清晰定义变量并检查单位。

10. Newton’s Law of Cooling and Proportional Change | 冷却定律与比例变化模型

Newton’s Law of Cooling is dT/dt = −k(T − Tₑ), where T is the object’s temperature and Tₑ is the constant ambient temperature. The solution is T(t) = Tₑ + (T₀ − Tₑ)e⁻ᵏᵗ. This is a separable linear DE.

牛顿冷却定律为 dT/dt = −k(T − Tₑ)，其中 T 为物体温度，Tₑ 为恒定环境温度。其解为 T(t) = Tₑ + (T₀ − Tₑ)e⁻ᵏᵗ。这是一个可分离变量的一阶线性微分方程。

In both IB and WJEC, you may also encounter proportional change models: the rate of change of a quantity is proportional to the difference between it and a limiting value (e.g., drug concentration in the bloodstream). Treat these using the separation method or integrating factor as appropriate.

在 IB 和 WJEC 中，还会遇到比例变化模型：某个量的变化率正比于该量与极限值的差（例如血液中的药物浓度）。根据题目类型，可采用分离变量法或积分因子法求解。

11. Coupled and Mixing Problems | 混合问题与方程耦合

Mixing problems involve a tank with inflow and outflow of a salt solution. Let A(t) be the amount of salt at time t. The DE is dA/dt = (rate in) − (rate out), where rate in = (concentration in)×(flow in) and rate out = (A(t)·flow out)/(volume at time t). If volume is constant, the equation is linear; if volume changes linearly, it may be linear or separable.

混合问题通常涉及一个容器的盐溶液流入和流出。设 A(t) 为 t 时刻盐的量。微分方程为 dA/dt = 流入速率 − 流出速率，其中流入速率 = 流入浓度×流入流量，流出速率 = A(t)·流出流量 / t 时刻的体积。若体积恒定，方程为线性；若体积线性变化，方程可能为线性或可分离。

These problems are cross-examined by allowing volume to vary—you must express volume as V(t) = V₀ + (r_in − r_out)t. Then the DE becomes a linear first-order that can be solved using the integrating factor. Keep careful track of units and simplifications.

这类问题常通过让体积变化来增加难度——你需要将体积表达为 V(t) = V₀ + (r_in − r_out)t。此时微分方程变为一阶线性方程，可利用积分因子求解。务必留心单位和化简过程。

12. Second-Order Homogeneous Linear Equations with Constant Coefficients | 二阶常系数齐次线性方程（进阶考点）

Although primarily a Further Mathematics topic, second-order DEs in the form ay” + by’ + cy = 0 occasionally appear as extension in IB HL papers or WJEC Mechanics contexts. The auxiliary equation ar² + br + c = 0 gives roots r₁, r₂. For real distinct roots, y = Ae^(r₁x) + Be^(r₂x); for repeated roots, y = (A + Bx)e^(rx); for complex roots α ± iβ, y = e^(αx)(A cos βx + B sin βx).

二阶方程虽主要属于进阶数学范畴，但形如 ay” + by’ + cy = 0 的常系数齐次线性方程偶尔会作为 IB HL 的拓展或出现在 WJEC 的力学情境中。特征方程 ar² + br + c = 0 的根为 r₁, r₂。若为两个不等实根，则通解为 y = Ae^(r₁x) + Be^(r₂x)；重根时为 y = (A + Bx)e^(rx)；共轭复根 α ± iβ 时为 y = e^(αx)(A cos βx + B sin βx)。

Knowing this structure helps in interpreting solutions of coupled first-order systems and understanding simple harmonic motion derived from Newton’s Second Law. For IB, you are not required to solve such equations from scratch unless explicitly taught, but recognising the form can give you an edge.

了解这一结构有助于理解一阶耦合系统的解以及由牛顿第二定律导出的简谐运动。对于 IB 课程，除非专门讲授，否则不要求从零求解此类方程，但识别其形式能让你在解题中占得先机。

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Differential Equations Exam Essentials | 微分方程考点精讲