IB Math Review Set 20C: Differential Equations | IB数学复习题集20C：微分方程

📚 IB Math Review Set 20C: Differential Equations | IB数学复习题集20C：微分方程

In the IB Mathematics: Analysis and Approaches HL course, Review Set 20C provides a focused collection of problems on differential equations, from separation of variables to real-world modelling. This article revisits the key skills assessed in the set, explains core concepts with bilingual notes, and offers step-by-step strategies for tackling typical exam-style questions. Whether you are preparing for a class test or the final examinations, mastering Review Set 20C will strengthen your ability to interpret, set up, and solve differential equations in contextual problems.

在IB数学：分析与方法HL课程中，复习题集20C集中考查了微分方程相关内容，涵盖分离变量法到实际建模。本文重温该题集所评估的核心技能，用中英双语解析核心概念，并提供应对典型考试题型的逐步策略。无论你是在准备课堂测验还是最终考试，掌握复习题集20C都将提升你理解、建立并求解应用题中微分方程的能力。

1. Overview of Review Set 20C | 复习题集20C概览

Review Set 20C in the Haese Mathematics HL textbook is the third review exercise for Chapter 20 (Differential Equations). It includes short-response and extended-response questions that test students' ability to classify differential equations, sketch slope fields, solve separable equations, and apply models such as exponential growth, logistic growth, and Newton's law of cooling. The set also prepares learners for the type of analytical reasoning required in Paper 1 and Paper 2.

Haese数学HL教材中的复习题集20C是第20章（微分方程）的第三个复习练习。它包含简答题和拓展题，考查学生分类微分方程、描绘斜率场、求解可分离变量方程以及应用指数增长、逻辑斯谛增长和牛顿冷却定律等模型的能力。该题集也为学习者在试卷一和试卷二中所需的推理论证能力打下基础。

2. What is a Differential Equation? | 什么是微分方程？

r of a differential equation is the highest derivative present; the degree refers to the power of the highest derivative when the equation is polynomial in derivatives. You need to recognise both linear and non-linear first-order equations.

微分方程是包含未知函数及其导数的方程。在IB HL课程中，学习的一阶方程通常采用dy/dx = f(x) g(y)（可分离变量）或dy/dx = f(x,y)的形式。微分方程的阶是指出现的最高阶导数；次是指在关于导数为多项式时最高阶导数的次数。你需要识别一阶线性与非线性方程。

General form: st-order). However, the IB syllabus emphasises separable equations and their applications.

一般形式：dy/dx + P(x)y = Q(x)（一阶线性）。但IB教学大纲更侧重可分离变量方程及其应用。

3. Slope Fields and Visualizing Solutions | 斜率场与解的可视化

lane, a short line segment is drawn with slope f(x,y). This allows you to sketch approximate solution curves without solving the equation analytically. In Review Set 20C, you may be asked to match slope fields with differential equations or to draw a solution curve passing through a given initial point.

斜率场（或方向场）是一阶微分方程dy/dx = f(x,y)的图形表示。在平面上每一点(x,y)处，画出斜率为f(x,y)的短线段。这样无需解析求解即可绘出近似解曲线。在复习题集20C中，你可能需要将斜率场与微分方程进行匹配，或画出经过给定初始点的解曲线。

Key observations:

关键观察：

• If f(x,y) depends only on y, the slope field has horizontal isoclines (lines of constant slope).
• 如果f(x,y)只依赖于y，斜率场会有水平等斜线（斜率恒定的直线）。
• Equilibrium solutions occur where dy/dx = 0, corresponding to constant functions.
• 平衡解发生在dy/dx = 0处，对应常值函数。

4. Separable Differential Equations | 可分离变量的微分方程

hen integrating both sides. Review Set 20C contains several equations that require careful algebraic manipulation before integration.

如果一个微分方程可写成dy/dx = g(x)h(y)的形式，则它是可分离变量的。分离变量法先将方程改写为(1/h(y)) dy = g(x) dx，再对两边积分。复习题集20C中有若干方程需要先进行仔细的代数变形，再积分。

Example: Solve dy/dx = 2xy, given y(0) = 3.
Separate: (1/y) dy = 2x dx → ∫ (1/y) dy = ∫ 2x dx → ln|y| = x² + C → y = A e^x². Using y(0)=3 gives A=3, so y = 3e^x².

例题：求解dy/dx = 2xy，满足y(0)=3。
分离变量：(1/y) dy = 2x dx → ∫ (1/y) dy = ∫ 2x dx → ln|y| = x² + C → y = A e^x²。利用y(0)=3得A=3，因此y = 3e^x²。

5. Solving Initial Value Problems | 求解初值问题

y₀. After finding the general solution (including an arbitrary constant), the initial condition is substituted to find the particular solution. Review Set 20C includes IVPs that lead to both explicit and implicit solutions— ed to leave an answer in implicit form when an explicit form is impossible or impractical.

初值问题（IVP）包含一个微分方程以及初始条件y(x₀) = y₀。在求得通解（含任意常数）后，代入初始条件以确定特解。复习题集20C包含导致显式解和隐式解的初值问题——当无法或不便于写出显式解时，要学会以隐式形式给出答案。

Tip: Always check that the solution satisfies the original differential equation. This verification can help you spot integration or algebraic errors.

提示：务必验证解是否满足原微分方程。这一检验能帮助你发现积分或代数错误。

6. Exponential Growth and Decay | 指数增长与衰减

tudies, radioactive decay, and continuously compounded interest. In Review Set 20C, you might need to determine the growth constant from given data or predict future values.

模型dy/dt = ky（k为常数）描述指数增长（k > 0）或衰减（k < 0）。其解为y = y₀ eᵏᵗ。这出现在种群研究、放射性衰变和连续复利计算中。在复习题集20C中，你可能需要根据给定数据确定增长常数，或预测未来值。

Half-life/doubling time: For decay, half-life T = (ln 2)/|k|; for growth, doubling time = (ln 2)/k.

广告 / Sponsored

半衰期/倍增时间：对于衰减，半衰期T = (ln 2)/|k|；对于增长，倍增时间 = (ln 2)/k。

7. Logistic Growth Model | 逻辑斯谛增长模型

lutions y=0 and y=L, and interpret the point of inflection where growth switches from increasing to decreasing.

逻辑斯谛微分方程dy/dt = ky(1 - y/L)引入了环境容纳量L。其解为S形曲线：y = L / (1 + A e⁻ᵏᵗ)，其中A由初始条件决定。复习题集20C要求学生分析逻辑斯谛方程的斜率场、识别平衡解y=0和y=L，并解释增长由递增转为递减的拐点。

Inflection point: Occurs at y = L/2, where the growth rate is maximum.

拐点：发生在y = L/2处，此时增长速率达到最大。

8. Newton's Law of Cooling | 牛顿冷却定律

Newton' Tₐ + (T₀ - Tₐ)e⁻ᵏᵗ. Review Set 20C problems often involve finding the cooling constant k from two temperature readings and then predicting the time needed to reach a certain temperature.

牛顿冷却定律指出，物体温度T的变化率与其温度和环境温度Tₐ之差成正比：dT/dt = -k(T - Tₐ)。这一可分离变量的方程给出T = Tₐ + (T₀ - Tₐ)e⁻ᵏᵗ。复习题集20C中常见的问题是，通过两次温度读数求出冷却常数k，再预测达到某温度所需的时间。

Practical note: Ensure time units are consistent when substituting temperature data.

实用要点：代入温度数据时需确保时间单位一致。

9. Application in Mixing Problems | 混合问题应用

These problems are excellent for testing modelling skills and appear in the extended-response sections of Review Set 20C.

混合问题涉及一个装有溶解物质（如盐）液体的容器。液体以一定速率流入和流出，目标是求出物质含量随时间变化的函数。微分方程基于净变化率建立：dA/dt = (流入速率) - (流出速率)。这类问题极好地考查建模能力，并出现在复习题集20C的拓展题中。

Typical form: at the same rate, then dA/dt = r cᵢ - (r/V₀) A, leading to a linear first-order equation.

典型形式：若浓度为cᵢ的盐水以速率r流入，且充分混合后以相同速率流出，则dA/dt = r cᵢ - (r/V₀) A，化为一阶线性方程。

10. Exam-Style Questions & Strategies | 考试题型与策略

In IB exams, differential equations questions are often structured to guide you through separation, substitution, integration, and interpretation. You may need to manipulate a given derivative into a separable form, recognise the need for partial fractions, or use integration by substitution. Review Set 20C mirrors this style, with part (a) asking for a slope field sketch, part (b) asking for the general solution, and part (c) applying an initial condition.

在IB考试中，微分方程题目通常分步引导你完成分离、代入、积分和解释。你可能需要将给出的导数化为可分离形式、识别需要使用部分分式，或运用代入积分法。复习题集20C仿照这一风格：第(a)部分要求描绘斜率场，第(b)部分求通解，第(c)部分应用初始条件。

Strategies:

解题策略：

• Always write the differential equation in standard form before planning your approach.
• 始终先将微分方程写成标准形式，再规划解题方案。
• When integrating, remember to include the constant of integration immediately.
• 积分时务必立即加上积分常数。
• Check the domain of the solution, especially if y appears in a denominator or inside a logarithm.
• 检查解的定义域，特别是当y出现在分母或对数中时。

11. Common Mistakes to Avoid | 常见错误及避免

Many errors in differential equations stem from premature simplification or overlooking absolute values. For instance, when integrating 1/y, the result is ln|y|, and ignoring the absolute value can lead to missing valid solution branches. Another frequent mistake is misapplying the initial condition before finding the general solution. Review Set 20C highlights these pitfalls through distractors in multiple-choice questions and follow-up reasoning tasks.

微分方程中的许多错误源于过早简化或忽略绝对值。例如，对1/y积分得ln|y|，忽视绝对值可能导致漏掉有效解分支。另一个常见错误是在求出通解之前错误代入初始条件。复习题集20C通过选择题干扰项和后续推理任务突显了这些容易出错的点。

Checklist:

检查清单：

• Did I separate variables correctly? Are all y-dependent terms on the same side?
• 我是否正确分离了变量？所有含y的项是否已集中到同一侧？
• Did I handle the chain rule correctly when substituting?
• 在换元时是否正处理了链式法则？
• Have I verified the solution satisfies the given equation?
• 我是否验证了解满足原方程？

12. Conclusion and Further Practice | 总结与拓展练习

Review Set 20C is an essential checkpoint for any IB HL student studying differential equations. By revisiting separable equations, slope fields, and contextual models such as logistic growth and cooling, you build a robust foundation for the final exam. After completing the set, test yourself again with past paper questions and try constructing your own differential equations from word problems to deepen your modelling intuition.

复习题集20C是每位学习微分方程的IB HL学生必经的重要关隘。通过重温可分离变量方程、斜率场以及逻辑斯谛增长和冷却等实际模型，你为最终考试打下了坚实基础。完成该题集后，用历年考题再次自测，并尝试根据文字问题自己建立微分方程，以加深建模直觉。

Remember that differential equations describe changing phenomena—treat each problem as a story to be translated into mathematics. Continual practice with Review Set 20C material will make that translation second nature.

记住，微分方程描述变化的现象——把每个问题当作一个故事，将其翻译成数学语言。不断练习复习题集20C中的材料，这种转换将成为你的第二天性。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）