📚 Solving First Order Differential Equations by Separation of Variables | 用分离变量法求解一阶微分方程
Differential equations are equations that involve an unknown function and its derivatives. In A‑Level Edexcel Mathematics, one of the most fundamental techniques for solving first‑order ordinary differential equations is the method of separation of variables. This approach allows us to rewrite the equation so that each variable appears exclusively on one side before integrating. Mastering this skill is essential for tackling both theoretical problems and real‑world modelling scenarios across the syllabus.
微分方程是包含未知函数及其导数的方程。在 A‑Level Edexcel 数学中，求解一阶常微分方程最基本的方法之一就是分离变量法。这种方法允许我们将方程重新整理，使每个变量只出现在等式的一边，然后进行积分。掌握这项技能对于应对考纲中的理论问题和实际建模情境都至关重要。
1. What is a First‑Order Differential Equation? | 什么是一阶微分方程？
A first‑ erivatives like d²y/dx². The general form can often be expressed as dy/dx = f(x, y), where f(x, y) is some function of x and y.
一阶微分方程是只包含一阶导数 dy/dx 以及可能包含变量 x 和 y 的方程。它不包含更高阶的导数，如 d²y/dx²。其一般形式通常可以表示为 dy/dx = f(x, y)，其中 f(x, y) 是 x 和 y 的某个函数。
A differential equation is called separable , the equation can be split so that all y terms are on the same side as dy and all x terms are on the same side as dx.
如果函数 f(x, y) 可以表示为仅含 x 的函数与仅含 y 的函数的乘积，即 dy/dx = g(x) × h(y)，则称该微分方程为可分离的。这时，我们可以将方程拆分，使得所有含 y 的项与 dy 在同一侧，所有含 x 的项与 dx 在同一侧。
2. The Core Principle of Separation | 分离变量的核心原理
g(x) dx, assuming h(y) ≠ 0. Both sides can now be integrated independently with respect to their own variables.
该方法的原理基于将导数视为微分的商。若 dy/dx = g(x)h(y)，我们可以将其整理为 (1/h(y)) dy = g(x) dx（假设 h(y) ≠ 0）。此时两边就可以分别对各自的变量进行积分了。
ften can be rearranged to give an explicit solution y = F(x) plus an arbitrary constant C.
对两边积分得到 ∫ (1/h(y)) dy = ∫ g(x) dx。积分后，我们得到一个关于 x 和 y 的隐式关系，通常可以重新整理，得到含任意常数 C 的显式解 y = F(x) + C 或类似形式。
3. Step‑by‑Step Procedure | 分离变量法分步求解步骤
Step 1: ify whether the differential equation is separable. Write it in the form dy/dx = g(x)h(y).
步骤一：判断微分方程是否可分离，将其写成 dy/dx = g(x)h(y) 的形式。
Step 2: eft with dy, and all terms involving x appear on the right with dx: (1/h(y)) dy = g(x) dx.
步骤二：重新整理方程，使所有含 y 的项与 dy 一起放在左边，所有含 x 的项与 dx 一起放在右边： (1/h(y)) dy = g(x) dx。
Step 3: Integrate both sides. Don't forget the constant of integration. It is usually enough to add a single constant on one side (commonly the right side).
步骤三：对两边分别积分。不要忘记积分常数。通常只需在一边（一般是右边）加一个常数 C 即可。
Step 4: If an initial condition is given, substitute the values to find the particular value of C. This yields the particular solution.
步骤四：如果给出了初始条件，将值代入以求出 C 的具体数值，从而得到特解。
Step 5: Where possible, rearrange the result to express y explicitly in terms of x, or leave it as an implicit relationship if that is simpler.
步骤五：若有可能，整理结果，将 y 显式地表示为 x 的函数，或者保持隐式关系（如果这样更简洁）。
4. General Solutions: The Role of the Arbitrary Constant | 通解：任意常数的作用
When we integrate without limits, we introduce an arbitrary constant C. The resulting family of curves is called the general solution. Each different value of C gives a different curve that satisfies the original differential equation.
当我们进行不定积分时，会引入任意常数 C。由此得到的一族曲线称为通解。C 的每一个不同取值都会给出满足原微分方程的一条不同曲线。
or y² = 2x² + 2C. This defines a family of hyperbolas, representing the general solution.
例如，考虑方程 dy/dx = 2x/y。分离变量得到 y dy = 2x dx。积分得 ½ y² = x² + C，或 y² = 2x² + 2C。这定义了一族双曲线，即通解。
5. Particular Solutions and Initial Conditions | 特解与初始条件
A particular solution is obtained when we are given extra information, typically a point (x₀, y₀) that the solution curve must pass through. This condition allows us to determine the constant C uniquely.
当题目给出了额外信息，通常是解曲线必须经过的某一点 (x₀, y₀) 时，我们就可以得到特解。这个条件使我们能够唯一确定常数 C。
16. Thus the particular solution is y² = 2x² + 16, and taking the positive square root y = √2x² + 16 because the initial y is positive.
例如，如果上例中附加条件 y(0) = 4，我们将 x = 0, y = 4 代入 y² = 2x² + K（K = 2C），得 16 = 0 + K，故 K = 16。于是特解为 y² = 2x² + 16，又因初始 y 为正，取正值可得 y = √2x² + 16。
6. Handling Logarithmic Integrals and Exponential Solutions | 处理对数积分与指数解
logarithms. For example, dy/dx = ky is separable as (1/y) dy = k dx. Integrating gives ln|y| = kx + C. Exponentiating then yields y = Aekx, where A = ±eC.
许多可分离方程包含 1/y 这类函数，积分后会出现自然对数。例如 dy/dx = ky 可分离为 (1/y) dy = k dx。积分得 ln|y| = kx + C。两边取指数后得到 y = Aekx，其中 A = ±eC。
It is important to remember to include the absolute value inside the logarithm, though when initial conditions determine a specific sign, the absolute value can usually be dropped safely at the end.
重要的是要记住在对数内加绝对值，虽然当初始条件确定了具体的符号后，通常可以在最后安全地去掉绝对值符号。
7. Verifying Solutions | 验证解的正确性
After finding a solution, it is good practice to verify it by differentiating and substituting back into the original differential equation. If the left‑hand side equals the right‑hand side, the solution is correct.
求出解之后，最好通过求导并代回原微分方程来进行验证。如果左边等于右边，那么解就是正确的。
hat y² = x² + 4 is a solution to dy/dx = x/y, we differentiate implicitly: 2y (dy/dx) = 2x ⇒ dy/dx = x/y. This matches the original equation, confirming our result.
例如，如果我们声称 y² = x² + 4 是 dy/dx = x/y 的解，我们隐式求导得：2y (dy/dx) = 2x ⇒ dy/dx = x/y。这与原方程匹配，从而验证了我们的结果。
8. Modelling with Separable Differential Equations | 利用分离变量法进行建模
In the Edexcel specification, separable differential equations appear frequently in contexts such as population growth, radioactive decay, Newton's Law of Cooling, and mixing problems. Recognising the structure of the differential equation derived from a word problem is a key skill.
在 Edexcel 考纲中，可分离的微分方程经常出现在人口增长、放射性衰变、牛顿冷却定律以及混合问题等情境中。能够识别由文字题导出的微分方程的结构是一项关键技能。
t size: dP/dt = kP. Separation gives dP/P = k dt, leading to the exponential model P = P₀ekt. Understanding the meaning of each constant (P₀ initial population, k growth rate) is vital for interpreting answers in exams.
例如，某种群 P 的增长速率与其当前规模成正比：dP/dt = kP。分离变量得 dP/P = k dt，进而得到指数模型 P = P₀ekt。理解每个常数的含义（P₀ 为初始种群数量，k 为增长率）对于在考试中解释答案至关重要。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Forgetting the constant of integration is one of the most frequent errors. Always add +C immediately after performing an indefinite integral, and then use initial conditions to find it.
忘记积分常数是最常见的错误之一。务必在每次进行不定积分后立即加上 +C，然后利用初始条件求出其值。
Incorrect separation in the form dy/dx = g(x)h(y). If it is dy/dx = x + y, separation is not directly possible.
分离不正确可能发生在学生将求和式误当作乘积的情形。请记住，方程必须能写成 dy/dx = g(x)h(y) 的形式。如果是 dy/dx = x + y，则不能直接分离。
Sign errors when exponentiating logarithmic equations are common. When you have ln|y| = kx + C, remember that ekx + C = eC · ekx, and eC is a positive constant, often rewritten as A. Don't drop the absolute value prematurely unless the context guarantees positivity.
对数方程取指数时的符号错误也很常见。当得到 ln|y| = kx + C 时，记住 ekx + C = eC · ekx，其中 eC 是一个正常数，通常重新记为 A。除非情境能确保变量恒为正，否则不要过早丢弃绝对值符号。
10. Exam Tips and Summary | 考试技巧与总结
In Edexcel past papers, marks are awarded for clear working, correct separation of variables, accurate integration, and correct evaluation of the constant. Even if you make a small slip in the final algebra, you can still gain method marks if your separation and integration steps are clearly set out.
在 Edexcel 历年真题中，清晰的解题过程、正确的变量分离、准确的积分以及常数的正确求解都会得到相应的分数。即便在最后的代数运算中出现小差错，只要分离变量和积分步骤写得清晰明了，仍可获得方法分。
Always check that your final solution actually satisfies the given differential equation. If time permits, differentiate your answer and plug it back into the original relationship. This quick verification can help you catch algebraic mistakes and secure full marks.
务必检查最终解是否真的满足给定的微分方程。如果时间允许，将你的答案求导并代回原等式。这种快速验证能帮助你发现代数错误，确保拿到满分。
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