Exercise H.6: Differential Equations | 微分方程练习 H.6

📚 Exercise H.6: Differential Equations | 微分方程练习 H.6

Exercise H.6 provides a focused set of problems designed to build fluency in solving first-order differential equations, particularly those that can be tackled by separating variables or by recognising standard forms. Mastering these exercises is essential for IB Mathematics: Analysis and Approaches Higher Level students, as differential equations form a bridge between pure calculus and real-world modelling. The problems in this set gradually increase in complexity, moving from straightforward separable equations to those requiring more careful algebraic manipulation and the application of initial conditions.

练习 H.6 提供了一组针对性的问题，旨在培养求解一阶微分方程的熟练度，尤其是那些可以通过分离变量或识别标准形式来解决的方程。掌握这些练习对于 IB 数学分析与方法高水平课程的学生至关重要，因为微分方程构成了纯微积分与现实世界建模之间的桥梁。这组问题的复杂性逐步增加，从简单的可分离方程过渡到需要更仔细的代数操作和应用初始条件的方程。

1. Recognising Separable First-Order Differential Equations | 识别可分离的一阶微分方程

e the right-hand side is the product of a function of x only and a function of y only. The key procedural step is to re-arrange the equation so that all terms involving y appear on the left-hand side with dy, and all terms involving x appear on the right-hand side with dx. After integration, the general solution often contains an arbitrary constant C, which can later be refined using initial or boundary conditions.

如果一个一阶微分方程可以写成 dy/dx = f(x)g(y) 的形式，即右边是仅含 x 的项与仅含 y 的项的乘积，则称其为可分离的。关键步骤是重新整理方程，使所有含 y 的项与 dy 一起出现在左边，所有含 x 的项与 dx 一起出现在右边。积分之后，通解通常包含一个任意常数 C，随后可以利用初始条件或边界条件加以确定。

• 步骤 1： 将方程改写为 g(y) dy = f(x) dx。
• 步骤 2： 两边同时积分：∫ g(y) dy = ∫ f(x) dx。
• 步骤 3： 如果可能，将 y 显式表达为 x 的函数。
• 步骤 4： 利用条件 (x₀, y₀) 求解常量 C。

• Step 1: Rewrite the equation as g(y) dy = f(x) dx.
• Step 2: Integrate both sides: ∫ g(y) dy = ∫ f(x) dx.
• Step 3: If possible, express y explicitly as a function of x.
• Step 4: Use the condition (x₀, y₀) to solve for the constant C.

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2. Worked Example: Basic Separation of Variables | 范例：基本分离变量法

x-terms to the right: (1/y) dy = 3x² dx, provided y ≠ 0. Integrate both sides to obtain ln |y| = x³ + C. Exponentiating both sides gives y = ±e^x³+C = A e^x³, where A = ±e^C onstant. Using the condition y(0) = 2, we find A = 2, so the particular solution is y = 2e^x³.

考虑微分方程 dy/dx = 3x²y，初始条件为 y(0) = 2。首先，通过将所有含 y 的项移到左边、含 x 的项移到右边来分离变量：(1/y) dy = 3x² dx，前提是 y ≠ 0。两边积分得到 ln |y| = x³ + C。对两边取指数得到 y = ±e^x³+C = A e^x³，其中 A = ±e^C 是一个新的任意常数。利用条件 y(0) = 2，求得 A = 2，因此特解为 y = 2e^x³。

sfies the original differential equation and the initial condition. Differentiating y = 2e^x³ gives dy/dx = 2e^x³ · 3x² = 3x²(2e^x³ . The domain of this solution is all real x, and the solution curve passes through (0, 2).

务必检查你的解是否满足原微分方程和初始条件。对 y = 2e^x³ 求导可得 dy/dx = 2e^x³ · 3x² = 3x²(2e^x³) = 3x²y，确认正确。该解的定义域为全体实数 x，且解曲线经过点 (0, 2)。

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3. Worked Example: Integrating Factor Method for Linear Equations | 范例：线性方程的积分因子法

rating both sides then yields the solution. For example, consider dy/dx + (1/x)y = x² for x > 0. Here P(x) = 1/x, so IF = e^{∫ 1/x dx} = e^{ln x} side is d/dx(xy), so d/dx(xy) = x³. Integrating gives xy = x⁴/4 + C, hence y = x³/4 + C/x.

并非所有一阶方程都是可分离的。形如 dy/dx + P(x)y = Q(x) 的一阶线性方程需要积分因子 IF = e^(∫ P(x) dx)。将整个方程乘以积分因子，可将左边转化为某个乘积的导数：d/dx(y · IF) = Q(x) · IF。然后对两边积分即可得解。例如，考虑方程 dy/dx + (1/x)y = x²，其中 x > 0。此时 P(x) = 1/x，因此 IF = e^{∫ 1/x dx} = e^{ln x} = x。两边乘以 x：x dy/dx + y = x³。左边正是 d/dx(xy)，因此 d/dx(xy) = x³。积分得 xy = x⁴/4 + C，故 y = x³/4 + C/x。

The integrating factor method is powerful because it always works for linear first-order ODEs, provided the integrals involved can be evaluated. Students often confuse this procedure with separation of variables; the key distinction is that separation requires the equation to be multiplicatively separable, whereas the linear form is additive in structure.

积分因子法功能强大，因为它对一阶线性常微分方程总是有效，只要所涉及的积分可以计算。学生常将此步骤与分离变量混淆；关键区别在于，分离变量要求方程在乘法上可分离，而线性形式在结构上是加性的。

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4. Applying Initial Conditions to Find Particular Solutions | 应用初始条件求特解

alue of C uniquely. This yields a particular solution rather than a family of curves. In the IB exam, you are almost always required to find the particular solution after a prompt such as 'given that y = 3 when x = 0, find y in terms of x'.

积分微分方程后，结果包含一个任意常数 C。初始条件或边界条件，通常写作 y(x₀) = y₀，使我们能够唯一确定 C 的值，从而得到特解而非一族曲线。在 IB 考试中，几乎总是要求你在给出诸如“已知当 x = 0 时 y = 3，求 y 关于 x 的表达式”这样的提示后求出特解。

π/4) = 0. Separating variables gives (1/(y²+1)) dy = tan x dx. Integrating: arctan y = -ln |cos x| + C. Using the condition: arctan 0 = -ln(√2/2) + C, so C = ln(√2/2) = -½ ln 2. The explicit solution is arctan y = -ln |cos x| - ½ ln 2, or equivalently, y = tan(-ln |cos x| - ½ ln 2).

考虑方程 dy/dx = (y² + 1) tan x，条件为 y(π/4) = 0。分离变量得 (1/(y²+1)) dy = tan x dx。积分：arctan y = -ln |cos x| + C。代入条件：arctan 0 = -ln(√2/2) + C，因此 C = ln(√2/2) = -½ ln 2。显式解为 arctan y = -ln |cos x| - ½ ln 2，或等价地 y = tan(-ln |cos x| - ½ ln 2)。

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5. Handling Logarithmic and Exponential Solutions | 处理含对数和指数的解

term is of the form 1/(ay + b). When integrating, remember that ∫ (1/(ay+b)) dy = (1/a) ln |ay + b| value can later be dropped. After obtaining a logarithmic expression, students frequently forget to exponentiate correctly or mishandle the constant C that emerges from the exponentiation step.

许多可分离方程在积分过程中会引出对数函数，特别是当分离的 y 项形如 1/(ay + b) 时。积分时请记住，∫ (1/(ay+b)) dy = (1/a) ln |ay + b|。在对数内部包含绝对值至关重要，尽管在许多应用问题中，量的符号是已知的，因此稍后可以去掉绝对值。在得到对数表达式后，学生经常忘记正确地进行指数运算，或者处理不当指数化步骤中出现的常数 C。

A typical mistake is writing e^ln|y|+C = y + e^C, instead of correctly simplifying to |y| = e^C · e^x² or y = A e^x² where A = ±e^C. Remember that e^a+b = e^a · e^b, not e^a + e^b. Practice simplifying these exponential expressions to avoid algebraic slips that can cost marks in an otherwise correct method.

一个典型错误是将 e^ln|y|+C 写成 y + e^C，而正确的简化应为 |y| = e^C · e^x² 或 y = A e^x²，其中 A = ±e^C。请记住 e^a+b = e^a · e^b，而不是 e^a + e^b。练习简化这些指数表达式，以避免在原本方法正确的情况下因代数失误而丢分。

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6. Verifying Solutions by Differentiation | 通过求导验证解

Once a candidate solution y(x) is found, you can always verify it by differentiating and substituting back into the original differential equation. This check is fast and provides confidence, especially when the algebraic manipulation was lengthy. To verify, compute dy/dx from your proposed solution, then simplify the right-hand side of the original equation using your expression for y. If both sides match identically, the solution is correct.

一旦找到了候选解 y(x)，你总是可以通过求导并代回原微分方程来验证。这种检验快速且能增强信心，尤其是在代数操作冗长的情况下。要验证，先从你所提出的解计算 dy/dx，然后利用 y 的表达式化简原方程的右边。如果两边完全相同，则解正确。

ate. This habit of verification is particularly valuable in an IB exam context, where time pressure can lead to small errors; a quick check may reveal a slip in a constant or a sign.

例如，要验证 y = 1/(2 - x) 满足 dy/dx = y²，计算 dy/dx = (1)/(2 - x)² = y²。两者直接匹配。这种验证习惯在 IB 考试环境下特别有价值，因为时间压力可能导致小错误；快速检查可能发现常数或符号上的疏漏。

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7. Common Pitfalls in Exercise H.6 | 练习 H.6 中的常见陷阱

Three errors appear repeatedly in student work on Exercise H.6. First, forgetting the constant of integration C. Without C, you obtain only one of infinitely many solution curves, and you will not be able to satisfy a given initial condition. Second, misapplying the chain rule when differentiating implicitly defined solutions. Third, incorrectly separating variables when the equation is not actually separable, leading to a dead end or a meaningless integral.

学生在完成练习 H.6 时常反复出现三类错误。第一，忘记积分常数 C。如果没有 C，你只能得到无穷多条解曲线中的一条，并且无法满足给定的初始条件。第二，在对隐式定义的解求导时错误地应用链式法则。第三，在方程实际上不可分离时错误地分离变量，导致走进死胡同或出现无意义的积分。

that should be noted or considered separately. In IB problems, you are usually expected to state any restrictions on the domain imposed by such divisions.

另一个微妙的陷阱涉及除以一个可能为零的表达式。例如，在分离 (y - 1) dy = x dx 时，除以 (y - 1) 隐含着 y ≠ 1 的假设。常数函数 y = 1 可能是一个奇异解，应予以注明或另行考虑。在 IB 题目中，通常要求你陈述由此类除法引起的定义域限制。

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8. Differential Equations with Trigonometric Functions | 含三角函数的微分方程

he solution will involve inverse trigonometric expressions, and it may be necessary to restrict the domain to ensure the function remains one-to-one. Pay attention to the periodic nature of trig functions when applying initial conditions, as multiple values may technically satisfy the condition but only one will be consistent with the selected branch.

涉及三角函数的方程，如 dy/dx = sin x · cos² y，需要仔细地对两边积分。当积分 1/cos² y = sec² y 时，回想其积分为 tan y。对于 x 一侧，∫ sin x dx = -cos x。解将涉及反三角表达式，并且可能需要限制定义域以确保函数保持一一对应。在应用初始条件时，注意三角函数的周期性，因为从技术上讲可能有多个值满足条件，但只有一个值与所选分支一致。

etermine that ½(π/4) + ¼ sin(π/2) = sin π + C, so π/8 + ¼ = 0 + C, giving C = π/8 + ¼. The solution remains implicit.

考虑方程 dy/dx = sec² y · cos x，条件为 y(π) = π/4。分离变量：cos² y dy = cos x dx（因为 1/sec² y = cos² y）。利用恒等式 cos² y = (1+cos 2y)/2 进行积分，得到 ½ y + ¼ sin 2y = sin x + C。这种隐式形式可能无法给出显式的 y(x)，但你仍然可以利用初始条件求 C。确定 ½(π/4) + ¼ sin(π/2) = sin π + C，因此 π/8 + ¼ = 0 + C，得到 C = π/8 + ¼。解仍保持隐式形式。

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9. Connecting to Real-World Modelling | 联系实际建模应用

ling in the IB syllabus. Exponential growth and decay models arise from dy/dt = ky, Newton' tive tools. Exercise H.6 may include context-rich problems where you must first formulate the differential equation from a verbal description and then solve it.

微分方程是 IB 教学大纲中数学建模的核心。指数增长和衰减模型源于 dy/dt = ky，牛顿冷却定律源于 dT/dt = -k(T - Tₐ)，逻辑斯谛增长源于 dP/dt = rP(1 - P/K)。这些方程都是可分离的，能提供强大的预测工具。练习 H.6 可能包含情境丰富的问题，要求你先根据文字描述建立微分方程，然后再求解。

When solving a modelling problem, clearly define your variables and state assumptions. For instance, in a cooling problem, assume the ambient temperature remains constant and the object's internal temperature gradient is negligible. Write the differential equation, solve it, apply the given data to find constants, and then answer the specific question — perhaps predicting the temperature at a specific time or the time to reach a threshold. Always interpret your mathematical result in the real-world context.

在求解建模问题时，要明确定义变量并陈述假设。例如，在冷却问题中，假设环境温度保持恒定且物体内部温度梯度可忽略。写出微分方程，求解，利用给定数据确定常数，然后回答具体问题——也许要求预测特定时刻的温度或达到某一阈值所需的时间。始终在现实世界的情境中解读你的数学结果。

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10. Summary and Exam Strategy for Exercise H.6 | 小结与备考策略

Success in Exercise H.6 rests on a systematic approach: identify the type of equation, apply the appropriate method (separation or integrating factor), integrate accurately, and use initial conditions to determine constants. Always write the final answer clearly, stating the domain if relevant. In exam settings, show all algebraic steps methodically, as the IB awards method marks even if the final answer contains a minor arithmetic slip. Double-check your solution through differentiation when time permits.

成功攻克练习 H.6 有赖于系统性方法：识别方程类型，应用适当的方法（分离变量或积分因子），准确积分，并利用初始条件确定常数。始终清晰地写出最终答案，如相关请注明定义域。在考试环境中，有条理地展示所有代数步骤，因为即使最终答案因小的算术失误而出错，IB 也会授予方法分。时间允许时，通过求导复查你的解。

Regular practice of a wide range of Exercise H.6 problems builds the fluency needed to recognise equation forms quickly and to avoid common traps. Aim to finish each problem with a verification step; this habit not only catches mistakes but also deepens your understanding of the relationship between a function and its derivative. With disciplined practice, differential equations become one of the most reliable and rewarding topics on the IB HL Mathematics paper.

经常性地、广泛地练习 H.6 中的各类题目，可以培养快速识别方程形式并避开常见陷阱的熟练度。力求在完成每道题时都加入验证步骤；这个习惯不仅能发现错误，还能加深你对函数与其导数之间关系的理解。通过有纪律的练习，微分方程将成为 IB 高水平数学试卷中最可靠且最有回报的专题之一。

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