Exercise 16D.1: Separable Differential Equations | 练习16D.1：可分离微分方程

📚 Exercise 16D.1: Separable Differential Equations | 练习16D.1：可分离微分方程

Differential equations that can be written so that all terms involving one variable appear on one side and all terms involving the other variable appear on the opposite side are called separable. Exercise 16D.1 is designed to build your fluency in recognising and solving this fundamental class of first-order ordinary differential equations. Mastering this technique is not only essential for the IB Mathematics: Analysis and Approaches and Applications and Interpretation syllabi but also forms the backbone for more advanced topics such as modelling population growth, radioactive decay, and Newton’s law of cooling.

能够被改写为一边只含有一个变量及其微分、另一边只含有另一个变量及其微分的微分方程叫作可分离微分方程。练习16D.1旨在培养你识别并求解这类一阶常微分方程的基本技能。无论是IB数学分析与方法还是应用与解释课程，掌握这个技巧都至关重要，它也是后续学习人口增长建模、放射性衰变和牛顿冷却定律等更深入主题的基础。

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1. Understanding Separable Differential Equations | 理解可分离微分方程

here g is a function of x alone and h is a function of y alone. The key idea is that the rate of change of y with respect to x can be factorised into a product of two independent functions. This structure allows us to rearrange the equation so that each variable appears exclusively with its own differential, leading to an integration problem on each side.

若一个一阶微分方程可以写成 dy/dx = g(x)·h(y) 的形式，其中 g 仅是 x 的函数，h 仅是 y 的函数，则称其为可分离方程。核心思想是 y 关于 x 的变化率可以被分解为两个独立函数的乘积。这样的结构允许我们将方程整理为每个变量仅与其自身微分共同出现的形式，从而将问题转化为两侧分别积分。

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2. The Standard Form | 标准形式

ient of two infinitesimal quantities and multiply both sides by dx. Then we divide by h(y)—provided h(y) ≠ 0— ds only on y and the right-hand side only on x, making the equation ready for integration.

我们需要先把给定的导数写为微分之商。如果 dy/dx = g(x)h(y)，可以把 dy/dx 看作两个无穷小量的商，并在等式两边同乘以 dx。接着除以 h(y)（假设 h(y) ≠ 0），得到 (1/h(y)) dy = g(x) dx。这一过程叫做分离变量。分离后的形式中，左侧只依赖于 y，右侧只依赖于 x，此时方程已为积分做好准备。

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3. Separation of Variables: Step-by-Step | 分离变量法：逐步解析

These must be considered separately because division by zero is not allowed during separation.
Move all factors containing y to the dy side and all factors containing x to the dx side.
Step 3: Verify that the equation is completely separated: no y terms remain on the x side, and no x terms remain on the y side.

第一步：识别常数解。检查 h(y)=0 是否会给出形如 y=常数的平衡解，这些解需单独考虑，因为在分离时不允许除以零。
第二步：将方程整理为 f(y) dy = g(x) dx。把所有含 y 的因子移到 dy 一侧，含 x 的因子移到 dx 一侧。
第三步：确认方程已完全分离：x 一侧无 y 残留，y 一侧无 x 残留。

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4. Integrating Both Sides | 两边积分

independently: ∫ f(y) dy = ∫ g(x) dx. Each integral is evaluated using standard techniques— ically added on only one side to avoid redundancy, but it is common practice to place it on the x side. The result is an implicit relationship between x and y that involves an arbitrary constant C.

变量分离后，我们对两侧独立积分：∫ f(y) dy = ∫ g(x) dx。每个积分使用标准方法计算——幂法则、三角积分、指数与对数法则，或在必要时使用部分分式。通常只在一侧添加积分常数以避免重复，惯例是将其加在 x 一侧。得到的结果是 x 与 y 之间包含任意常数 C 的隐式关系。

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5. Dealing with Constants of Integration | 处理积分常数

The constant of integration C arises from the indefinite integrals. After integration, we often manipulate the resulting expression to solve for y explicitly. It is important to remember that adding a constant on one side is sufficient because the difference of two arbitrary constants is still a single arbitrary constant. When exponentiating both sides, a constant inside a logarithm transforms into a multiplicative constant: e^{ln|y| + C} = e^C |y|, and e^C entionally replaced by a new positive constant A (or a constant that can absorb a ± sign).

积分常数 C 源于不定积分。积分后，我们常常对所得表达式进行变形，以得出 y 的显式解。需要记住只在一侧加常数就足够了，因为两个任意常数之差仍为一个任意常数。当两侧同时取指数时，对数内的常数会变为乘法常数：e^ln|y|+C = e^C |y|，习惯上会用新的正常数 A（或可吸收正负号的常数）来替换 e^C。

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6. Finding Particular Solutions | 求特解

n and solve for C. The resulting particular solution satisfies both the differential equation and the given condition. In IB problems, you are frequently asked to find a particular solution passing through a given point. Always check that the initial value does not coincide with a constant solution excluded earlier.

初始条件，例如 y(x₀)=y₀，可以让我们确定任意常数 C 的值。求出通解后，将 x=x₀ 和 y=y₀ 代入方程并解出 C。所得特解同时满足微分方程和给定条件。在 IB 试题中，经常要求你求出通过某给定点的特解。务必检查初始值是否与之前排除的常数解重合。

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7. Implicit vs. Explicit Solutions | 隐式解与显式解

ntary functions. In such cases, leaving the solution in implicit form is perfectly acceptable and often expected. For example, the equation ln|y| t expression still defines y as a function of x, and it satisfies the original differential equation. The IB mark scheme accepts both forms unless the question explicitly requires an explicit form.

有时积分后的方程无法用初等函数解出 y。这时将解保留为隐式形式是完全可接受的，且经常是题目所期望的。例如，方程 ln|y|+y² = x³+C 无法通过代数变形为 y=f(x)。隐式表达式仍然将 y 定义为 x 的函数，且满足原微分方程。除非题目明确要求显式形式，否则 IB 阅卷标准接受两种写法。

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8. Common Mistakes to Avoid | 常见错误

One frequent error is forgetting to include the absolute value when integrating 1/y, which gives ln|y|, not ln y. Another is mishandling the constant: when exponentiating, students sometimes incorrectly write e^{ln y + C} = y + C instead of e^{ln y}·e^C = |y|e^C. Also, do not lose constant solutions by dividing by an expression that could be zero without checking. Finally, always verify that the separated integrals are set up correctly; misplacing a negative sign or missing a factor can lead to an entirely wrong solution curve.

一个常见错误是积分 1/y 时忘记加绝对值，正确结果是 ln|y| 而非 ln y。另一个常见问题是常数处理不当：取指数时，有些学生错误地写成 e^{ln y+C} = y+C，而正确形式应是 e^{ln y}·e^C = |y|e^C。此外，不要因除以可能为零的表达式而丢失常数解，务必事先检查。最后，务必验证分离后的积分形式是否正确；错放负号或遗漏因子都可能导致完全错误的解曲线。

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9. Worked Examples from Exercise 16D.1 | 练习16D.1例题解析

Example 1: Solve dy/dx = 3x²y.
Separate: (1/y) dy = 3x² dx, y ≠ 0. Integrate: ln|y| = x³ + C. Exponentiate: |y| = e^x³+C = e^C e^x³. Thus y = A e^x³, where A = ±e^C (A ≠ 0). The constant solution y = 0 is also covered if we allow A = 0.

例1：解 dy/dx = 3x²y。
分离变量：(1/y) dy = 3x² dx，y ≠ 0。积分：ln|y| = x³ + C。取指数：|y| = e^x³+C = e^C e^x³。因此 y = A e^x³，其中 A = ±e^C (A ≠ 0)。若允许 A = 0，常数解 y = 0 也被包含在内。

Example 2: Solve dy/dx = (x + 1)/(y²), y(0) = 2.
= 0 + 0 + K → K = 16. Particular solution: 2y³ = 3x² + 6x + 16, or y = ((3/2)x² + 3x + 8)^1/3.

例2：解 dy/dx = (x+1)/(y²)，y(0)=2。
分离变量：y² dy = (x+1) dx。积分：(1/3)y³ = (1/2)x² + x + C。两边乘6：2y³ = 3x² + 6x + 6C。令 K=6C。代入初始条件：x=0, y=2 → 2(8)=0+0+K → K=16。特解：2y³ = 3x² + 6x + 16，或 y = ((3/2)x² + 3x + 8)^1/3。

Example 3: Solve (1+x) dy/dx = y(2−y), giving the general solution in implicit form.
= 1/(1+x) dx. Use partial fractions: 1/(y(2−y)) = 1/2 (1/y + 1/(2−y)). Integrate: (1/2)(ln|y| − ln|2−y|) = ln|1+x| + C. Multiply by 2: ln|y/(2−y)| = 2 ln|1+x| + 2C. Exponentiate: |y/(2−y)| = e^2C (1+x)². Let B = ±e^2C y) = B(1+x)². This is the implicit form; it can be rearranged to explicit form if desired.

例3：解 (1+x) dy/dx = y(2−y)，用隐式形式给出通解。
分离变量：1/(y(2−y)) dy = 1/(1+x) dx。用部分分式：1/(y(2−y)) = 1/2 (1/y + 1/(2−y))。积分：(1/2)(ln|y| − ln|2−y|) = ln|1+x| + C。乘2：ln|y/(2−y)| = 2 ln|1+x| + 2C。取指数：|y/(2−y)| = e^2C (1+x)²。令 B = ±e^2C，得 y/(2−y) = B(1+x)²。此为隐式形式；若需要可化为显式。

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10. Tips for Effective Practice | 高效练习提示

mmon separable structures such as dy/dx = k y, dy/dx = xy, dy/dx = (x²+1)/y, and dy/dx = eˣ heck by differentiating your answer and confirming that it satisfies the original equation—this is the fastest way to catch algebraic mistakes.

在完成练习 16D.1 时，首先判断方程是否已经可分离，或能否分解为 x 的函数与 y 的函数的乘积。多加练习以识别常见的可分离结构，如 dy/dx = k y、dy/dx = xy、dy/dx = (x²+1)/y 和 dy/dx = eˣ sin y。将分离变量的代数操作与积分步骤明确分开。求出解后，务必通过微分验证它是否满足原方程——这是发现代数错误的最快方法。

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11. Connecting to Slope Fields | 与斜率场的联系

, the slope field of dy/dx = x/y shows horizontal segments on the x-axis (where y=0 gives infinite slope, actually undefined) and vertical slopes along the y-axis. Recognising how the general solution aligns with the slope field deepens your understanding and helps verify that no solutions have been overlooked.

斜率场（方向场）为可分离微分方程提供了几何解释。在每一点 (x,y) 处，短小线段的斜率等于 g(x)h(y)。积分后得到的解曲线族应与场中显示的斜率保持一致。例如，dy/dx = x/y 的斜率场在 x 轴上呈现水平线段（y=0 时斜率无穷大，实则无定义），沿 y 轴斜率垂直。理解通解如何与斜率场对齐，能加深理解并帮助检验是否遗漏解。

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12. Summary and Key Takeaways | 总结与要点

ure involves separation, integration, handling the constant correctly, and applying initial conditions if provided. Always watch for constant solutions hidden in the division step. With consistent practice on exercises like 16D.1, you will be able to solve these equations quickly and accurately, setting a strong foundation for later topics such as homogeneous equations, integrating factors, and real-world modelling scenarios.

可分离微分方程是那些能够被整理为 f(y) dy = g(x) dx 的方程。求解步骤包括分离变量、积分、正确管理常数，以及在有需要时应用初始条件。务必留意在分离步骤中隐藏的常数解。通过像16D.1这样的习题不断练习，你将能够快速准确地求解这类方程，为齐次方程、积分因子以及实际建模等后续内容打下坚实基础。

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