Exercise 14G: Differential Equations – Solving by Separation of Variables | 练习14G：微分方程 – 分离变量法求解

📚 Exercise 14G: Differential Equations – Solving by Separation of Variables | 练习14G：微分方程 – 分离变量法求解

Exercise 14G focuses on one of the most powerful and fundamental techniques for solving first-order ordinary differential equations: separation of variables. This skill is essential for IB Mathematics Analysis & Approaches and Applications & Interpretation courses, particularly when modelling real-world phenomena such as population growth, radioactive decay, and cooling processes. In this article, we will walk through the key concepts, worked examples, common pitfalls, and practical tips to help you master this exercise set with confidence.

练习14G侧重于求解一阶常微分方程最强大、最基本的技巧之一：分离变量法。这项技能对IB数学分析与方法和应用与解释课程至关重要，尤其是在对现实世界现象（如人口增长、放射性衰变和冷却过程）进行建模时。在本文中，我们将逐步讲解关键概念、典型例题、常见错误以及实用技巧，帮助你自信地掌握这一组练习。

---

1. Understanding the Structure of Exercise 14G | 了解练习14G的结构

Exercise 14G typically presents a series of differential equations that can be rearranged so that all terms involving the dependent variable y appear on one side of the equation, while all terms involving the independent variable x appear on the other side. You are often asked to find the general solution, and in some problems, a particular solution is required using a given initial condition. The problems increase in complexity, starting from simple exponential forms to equations that require algebraic manipulation before separation.

练习14G通常会呈现一系列微分方程，这些方程可以重新排列，使得所有包含因变量y的项出现在方程的一边，所有包含自变量x的项出现在另一边。通常要求你求出通解，而在某些问题中，需要利用给定的初始条件求出特解。题目复杂性逐步增加，从简单的指数形式发展到需要先进行代数运算才能分离的方程。

---

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

ferential equation where the rate of change of y with respect to x is given as a function of x. In IB Mathematics, you will mainly deal with first-order separable equations, which are the foundation for understanding more advanced topics such as coupled differential equations or second-order linear equations in further study.

微分方程是包含未知函数及其导数的方程。例如，dy/dx = 3x² 是一个简单的一阶微分方程，其中y关于x的变化率被表示为一个关于x的函数。在IB数学中，你主要会接触一阶可分离方程，这是理解更高级主题（如耦合微分方程或二阶线性方程）的基础。

---

3. The Core Principle: Separation of Variables | 核心原理：分离变量法

x) dx. Integrating both sides then yields the general solution, usually involving an integration constant C. The key is to recognise when a differential equation is separable and to perform the algebraic steps carefully.

分离变量法的工作原理是，先将导数项 dy/dx 分离出来，然后通过代数乘法将两边变形，使得每个含有y的项都与 dy 在一起，每个含有x的项都与 dx 在一起。用符号表示，如果我们有 dy/dx = g(x)·h(y)，则将其重写为 (1/h(y)) dy = g(x) dx。然后对两边积分便得到通解，通常会包含一个积分常数C。关键在于识别一个微分方程何时是可分离的，并小心地执行代数步骤。

---

4. Example 1: A Basic Exponential Growth Model | 例1：基本指数增长模型

y. Rewriting gives (1/y) dy = 2 dx. Integrating both sides yields ∫(1/y) dy = ∫2 dx, so ln|y| = 2x + C. Exponentiating gives |y| = e^2x+C = e^C·e^2x. Let A = ±e^C, then the general solution is y = A e^2x, where A is an arbitrary constant. This form is typical for natural growth or decay processes where the rate of change is proportional to the quantity present.

考虑微分方程 dy/dx = 2y。这是可分离的，因为右边可以写成 2·y。重写后得到 (1/y) dy = 2 dx。两边积分得 ∫(1/y) dy = ∫2 dx，因此 ln|y| = 2x + C。取指数后得到 |y| = e^2x+C = e^C·e^2x。令 A = ±e^C，则通解为 y = A e^2x，其中A为任意常数。这种形式是自然增长或衰减过程的典型形式，其中变化率与当前数量成正比。

---

5. Example 2: Integrating Trigonometric Functions | 例2：三角函数的积分

equation dy/dx = y cos x. Separating variables: (1/y) dy = cos x dx. Integration gives ln|y| = sin x + C. Thus, y = A e^{sin x}, where A is a non-zero constant (or zero in the trivial solution). Notice that the integration step relies on recognising the integral of cos x as sin x. IB students must be fluent with standard integrals, especially polynomials, exponentials, logarithms, and trigonometric functions, as they frequently appear in separable equations.

求解方程 dy/dx = y cos x。分离变量：(1/y) dy = cos x dx。积分得 ln|y| = sin x + C。因此，y = A e^{sin x}，其中A是非零常数（或在平凡解中为零）。注意积分步骤依赖于识别 cos x 的积分是 sin x。IB学生必须熟练掌握标准积分，特别是多项式、指数、对数和三角函数，因为它们经常出现在可分离方程中。

---

6. Example 3: An Equation Requiring Algebraic Rearrangement | 例3：需要代数重排的方程

etimes the equation is not immediately recognisable as separable. For instance, dy/dx = (y – an still separate: (1/(y-1)) dy = (1/(x+2)) dx. Integrating: ∫(1/(y-1)) dy = ∫(1/(x+2)) dx ⇒ ln|y-1| = ln|x+2| + C. Using log properties, ln|y-1| = ln|A(x+2)| after letting C = ln|A|. Hence, y – onstrates that separation often works for rational functions where variables can be split.

有时方程并不立即呈现出可分离的形式。例如，dy/dx = (y – 1) / (x + 2)。这里我们仍然可以分离：(1/(y-1)) dy = (1/(x+2)) dx。积分得：∫(1/(y-1)) dy = ∫(1/(x+2)) dx ⇒ ln|y-1| = ln|x+2| + C。利用对数性质，令 C = ln|A|，则 ln|y-1| = ln|A(x+2)|。因此，y – 1 = A(x+2)，或 y = A(x+2) + 1。如果给定初始点，常数A便可确定。此例表明，对于可以将变量分开的有理函数，分离法通常都有效。

---

广告 / Sponsored

7. Working with Initial Conditions to Find Particular Solutions | 利用初始条件求特解

x and y values to solve for that constant. For example, if the general solution is y = A e^{sin x} and the condition is y(π/2) = 1, then 1 = A e^sin(π/2) = A e¹, so A = e^-1. The particular solution is y = e^-1 e^{sin x} = e^{sin x – 1}. Always simplify the final expression unless instructed otherwise.

练习14G中的许多问题都包含初始条件，如 y(0) = 5 或 y(π/2) = 3。在得到含有任意常数的通解后，代入给定的x和y值以求解该常数。例如，若通解为 y = A e^{sin x}，条件为 y(π/2) = 1，则 1 = A e^sin(π/2) = A e¹，故 A = e^-1。特解为 y = e^-1 e^{sin x} = e^{sin x – 1}。除非另有指示，总是要化简最终表达式。

---

8. Common Pitfalls and How to Avoid Them | 常见错误及如何避免

One frequent mistake is forgetting the absolute value inside the logarithm when integrating 1/y. While the constant of integration absorbs the sign in the final answer, it is good practice to write ln|y| to avoid losing valid solutions. Another error is misapplying the integration constant: adding it on only one side is sufficient; adding it to both sides can lead to confusion. Also, students sometimes attempt to separate variables before identifying that the equation is not separable—always check if the right-hand side can be factored into a product of a function of x and a function of y. Finally, be cautious when dividing by an expression that could be zero; this might exclude singular solutions.

一个常见错误是在积分 1/y 时忘记对数内部的绝对值。尽管积分常数在最终答案中吸收了符号，但最佳做法是写 ln|y|，以免丢失有效解。另一个错误是误加积分常数：只需在其中一边加上即可；两边都加会导致混淆。此外，学生有时在未确定方程是否可分离之前就贸然分离变量——务必先检查右边是否可分解为一个关于x的函数与一个关于y的函数的乘积。最后，在除以可能为零的表达式时要小心；这可能会排除奇异解。

---

9. A Quick Reference Table of Standard Separable Forms | 标准可分离形式速查表

The following table summarises some common separable differential equations and their general solutions. Use it as a mental checklist when working through Exercise 14G.

下表总结了一些常见的可分离微分方程及其通解。在练习14G中可将其作为思维检查表使用。

Differential Equation	Separated Form	General Solution
dy/dx = ky	(1/y) dy = k dx	y = A ekx
dy/dx = g(x)	dy = g(x) dx	y = ∫g(x)dx + C
dy/dx = f(y)	(1/f(y)) dy = dx	∫(1/f(y))dy = x + C
dy/dx = g(x)/h(y)	h(y) dy = g(x) dx	∫h(y)dy = ∫g(x)dx + C

---

10. Linking to IB Modelling Questions | 与IB建模题的联系

ext of real-life modelling. For instance, Newton’s law of cooling states that dT/dt = -k(T – arable. Population dynamics with limited resources may lead to equations like dP/dt = rP(1 – by ensuring you can handle algebraic separation and integration fluently, freeing up cognitive energy to interpret the context and set up the correct derivative relationship.

在IB考试中，可分离微分方程经常出现在现实建模的背景下。例如，牛顿冷却定律指出 dT/dt = -k(T – T_ambient)，这是可分离的。资源有限的人口动态可能产生形如 dP/dt = rP(1 – P/K) 的方程，分离后变为一个涉及部分分式的逻辑斯谛方程。练习14G的题目通过确保你能熟练处理代数分离和积分，为这些应用做好准备，从而使你能够腾出认知精力来解释背景并建立正确的导数关系。

---

11. Step-by-Step Strategy for Tackling Exercise 14G | 攻克练习14G的分步策略

1. Identify the form: check if dy/dx equals a product of f(x) and g(y), or can be rearranged into that form.
2. Multiply both sides to achieve the separated structure: (1/g(y)) dy = f(x) dx.
3. Integrate both sides, adding the constant of integration on the right side only (or the left—just one side).
4. Solve for y explicitly if possible, using algebraic manipulation and properties of exponentials/logarithms.
5. If an initial condition is given, substitute to find the particular constant, and simplify the final answer.
6. Check for any domain restrictions or lost solutions due to division by zero.

1. 识别形式：检查 dy/dx 是否等于 f(x) 与 g(y) 的乘积，或者是否可以重排成这种形式。
2. 两边同乘以达到分离结构：(1/g(y)) dy = f(x) dx。
3. 两边积分，只在右边（或左边——仅一侧）加上积分常数。
4. 如果可能，通过代数操作以及指数/对数性质显式解出 y。
5. 若给定初始条件，代入求特定常数，并化简最终答案。
6. 检查是否由于除以零而导致定义域限制或丢失解。

---

12. Conclusion and Further Practice | 结语与进一步练习

Mastering separation of variables is a cornerstone of the IB Mathematics differential equations topic. Exercise 14G offers a structured pathway to build this skill from basic to intermediate levels. After completing this exercise, you should feel comfortable recognising separable equations, performing the necessary algebraic steps, integrating standard functions, and applying initial conditions. For further consolidation, try creating your own separable equations from real-world scenarios or explore how separation extends to partial differential equations in future studies. Remember, consistent practice is the key to fluency.

掌握分离变量法是IB数学微分方程主题的基石。练习14G为你提供了一条从基础到中级构建这一技能的结构化路径。完成本练习后，你应该能够自如地识别可分离方程、执行必要的代数步骤、积分标准函数并应用初始条件。为了进一步巩固，尝试从现实场景创建自己的可分离方程，或在未来学习中探索分离变量法如何延伸到偏微分方程。请记住，持续不断的练习是达到熟练的关键。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）