📚 Solving Differential Equations by Separation of Variables | 用分离变量法解微分方程
Many real-world situations involve rates of change – population growth, radioactive decay, heat loss, or the spread of a disease – and these are modelled using differential equations. In A-Level Mathematics, one of the most fundamental and widely applied techniques for solving first-order ordinary differential equations is the method of separation of variables. This method allows you to rearrange an equation involving a derivative so that all terms containing one variable appear on one side of the equation, and all terms containing the other variable on the other side. Once separated, integration gives the general solution, and any given conditions allow you to determine the particular solution.
许多现实世界的情形都涉及变化率——人口增长、放射性衰变、热量散失或疾病传播——我们用微分方程来建模。在 A-Level 数学中，求解一阶常微分方程最基本且应用最广泛的方法之一就是分离变量法。通过这种方法，你可以重新排列含有导数的方程，使所有含一个变量的项出现在等式的一边，所有含另一个变量的项出现在另一边。一旦分离成功，积分就能给出通解，而任何给定的条件则能让你求出特解。
1. Recognising a Separable First-Order ODE | 识别可分离的一阶常微分方程
de can be expressed as a product of a function of x only and a function of y only, i.e. dy/dx = g(x) × h(y). For example, dy/dx = 3x²y is separable because it equals (3x²)(y). However, dy/dx = x + y is not separable as it cannot be written purely as a product of a function of x and a function of y.
一阶常微分方程通常可以写成 dy/dx = f(x, y) 的形式。如果右手边可以表示为仅含 x 的函数与仅含 y 的函数的乘积，即 dy/dx = g(x) × h(y)，那么这个方程就是可分离的。例如，dy/dx = 3x²y 是可分离的，因为它等于 (3x²)(y)。然而，dy/dx = x + y 不能写成纯乘积形式，因此不可分离。
2. Core Idea of Separation of Variables | 分离变量法的核心思想
the equation, and all x-dependent terms, including dx, move to the other side. This transforms the differential equation into an equality of two differentials: (1/h(y)) dy = g(x) dx. Both sides can then be integrated independently. The constants of integration are combined into a single constant C.
核心思想是把 dy/dx 当作一个分式，重新排列微分方程，使得所有依赖于 y 的项（包括 dy）移到等式的一边，所有依赖于 x 的项（包括 dx）移到另一边。这样一来，微分方程就变成了两个微分之间的等式：(1/h(y)) dy = g(x) dx。然后两边分别积分。积分常数合并成一个统一的常数 C。
3. Step-by-Step Procedure | 分步求解步骤
Step 1: Rewrite the derivative if necessary. Replace dy/dx with the differential form.
如有需要，重写导数。将 dy/dx 写成微分形式。
Step 2: Separate the variables. Multiply both sides by dx and divide by any factor that prevents a clean separation, ensuring all y-terms are with dy and all x-terms with dx.
分离变量。两边同时乘以 dx，并除以任何阻碍干净分离的因子，确保所有含 y 的项与 dy 在一起，所有含 x 的项与 dx 在一起。
Step 3: Integrate both sides. Add a single constant of integration C to one side (conventionally the right-hand side).
两边积分。在一边（通常是在右边）添加一个积分常数 C。
Step 4: Simplify to obtain the general solution, expressing y in terms of x if possible. If an initial condition is given, substitute to find C and write the particular solution.
化简得到通解，如果可能，将 y 表示成 x 的显函数。如果给定了初始条件，代入求出 C，写出特解。
4. Simple Example with Explicit Variables | 变量显式的简单示例
nd multiplying by dx: (1/y) dy = 2x dx. Now integrate: ∫ (1/y) dy = ∫ 2x dx. This gives ln|y| = x² + C. Exponentiating both sides yields |y| = ex² + C = eC · ex². Letting A = ±eC (where A is a non-zero constant) we obtain y = A ex². The general solution thus includes an arbitrary constant A.
考虑 dy/dx = 2xy。通过两边除以 y 并乘以 dx 来分离变量：(1/y) dy = 2x dx。然后积分：∫ (1/y) dy = ∫ 2x dx。得到 ln|y| = x² + C。两边取指数：|y| = ex² + C = eC · ex²。令 A = ±eC（A 为非零常数），我们得到 y = A ex²。通解因此包含一个任意常数 A。
5. Using Initial Conditions to Find a Particular Solution | 利用初始条件求特解
o told that y = 3 when x = 0, we substitute these values into the general solution y = A ex² to find A. Since e0² = 1, we have 3 = A × 1, so A = 3. The particular solution is y = 3 ex². Always check that the solution satisfies the original differential equation and the initial condition.
如果还告诉我们当 x = 0 时 y = 3，我们将这些值代入通解 y = A ex² 以求出 A。由于 e0² = 1，我们有 3 = A × 1，因此 A = 3。特解为 y = 3 ex²。始终检查解是否满足原微分方程和初始条件。
6. Dealing with Exponential Growth and Decay Models | 处理指数增长与衰减模型
wth (if k > 0) or decay (if k < 0). Separate to obtain (1/y) dy = k dt, then integrate: ln|y| = k t + C. Hence, y = A ek t initial values such as the initial population y₀ at time t = 0, giving A = y₀, so y = y₀ ek t rovided for decay, you can determine k using the condition y = ½ y₀ at the half-life time.
一个经典的可分离微分方程是 dy/dt = k y，其中 k 为常数。它描述了指数增长（若 k > 0）或衰减（若 k < 0）。分离变量得到 (1/y) dy = k dt，然后积分：ln|y| = k t + C。因此，y = A ek t。在许多考试题目中，你会得到初始值，例如在时刻 t = 0 时的初始数量 y₀，从而求出 A = y₀，于是 y = y₀ ek t。如果是衰减问题中提供了半衰期，你可以用半衰期时刻 y = ½ y₀ 这一条件来求出 k。
7. Newton's Law of Cooling Context | 牛顿冷却定律情景
T and the ambient temperature Tₐ. This gives the differential equation dT/dt = −k (T − Tₐ), where k > 0. The equation is separable: (1/(T − Tₐ)) dT = −k dt. Integration yields ln|T − Tₐ| = −k t + C, so T − Tₐ = A e−k t nitial temperature T₀ is known at t = 0, then A = T₀ − Tₐ, leading to T = Tₐ + (T₀ − Tₐ) e−k t. This model frequently appears in contextual problems.
牛顿冷却定律指出，物体温度的变化率与其自身温度 T 与环境温度 Tₐ 之差成正比。由此得到微分方程 dT/dt = −k (T − Tₐ)，其中 k > 0。该方程是可分离的：(1/(T − Tₐ)) dT = −k dt。积分后得到 ln|T − Tₐ| = −k t + C，于是 T − Tₐ = A e−k t。如果在 t = 0 时已知初始温度 T₀，则 A = T₀ − Tₐ，从而 T = Tₐ + (T₀ − Tₐ) e−k t。这一模型经常出现在应用题中。
8. Handling Equations Requiring Algebraic Rearrangement | 处理需要代数重排的方程
y + 2x can be written as dy/dx = x(y + 2). Now it is clearly separable. Another common form is (1 + x) dy/dx = y(2 − x). Here, divide both sides by y(1 + x) and multiply by dx to get (1/y) dy = (2 − x)/(1 + x) dx. Always simplify the fraction on the x-side before integrating, perhaps using polynomial division or partial fractions.
并非所有可分离方程都以简单的 dy/dx = g(x)h(y) 形式出现。有时你需要进行因式分解或重新排列。例如，dy/dx = xy + 2x 可以写成 dy/dx = x(y + 2)，此时显然可分离。另一种常见形式是 (1 + x) dy/dx = y(2 − x)。这里，两边除以 y(1 + x) 再乘以 dx，得到 (1/y) dy = (2 − x)/(1 + x) dx。在积分之前，始终要将 x 一侧的分式化简，或许要使用多项式除法或部分分式。
9. Integration Pitfalls and the Role of the Absolute Value | 积分陷阱与绝对值的作用
When integrating 1/y, the result is ln|y|, not ln(y). The absolute value is essential because the logarithm is only defined for positive arguments. When you later exponentiate to remove the log, you write eln|y| = |y|, and the constant eC emember that the integration constant C can represent any real number, but after exponentiation the constant A = ±eC is non-zero; you must check whether y = 0 is a solution separately.
对 1/y 积分时，结果是 ln|y|，而不是 ln(y)。绝对值必不可少，因为对数只对正的参数有定义。当你随后用指数消去对数时，写出 eln|y| = |y|，常数 eC 可以吸收符号。很多学生会忘记加模长而丢分。在常数被重新定义之前，请始终保留绝对值符号。还要记住，积分常数 C 可以表示任何实数，但取指数后常数 A = ±eC 不为零；你必须单独检查 y = 0 是否也是解。
10. Common Mistakes and How to Avoid Them | 常见错误及其避免方法
A frequent error is attempting to separate variables when the equation is not separable; always check the structure. Another mistake is forgetting to include the dx or dy after separation, which makes integration invalid. Some students integrate without a clear differential on each side. Also, when an initial condition is used, be sure to apply it to the general solution before any final simplification, as premature simplification can lead to algebraic errors. Finally, in applied problems, units and context often dictate whether a constant is positive or negative – interpret the sign of k carefully.
一个常见的错误是当方程实际上不可分离时却试图分离变量；务必先检查方程的结构。另一个错误是分离后忘记包含 dx 或 dy，这会使积分无效。有些学生在两边没有清晰的微分符号时就直接积分。此外，当使用初始条件时，一定要在进行最终化简之前先将其代入通解，过早化简可能导致代数错误。最后，在应用题中，单位和物理情景常常决定常数的正负——要仔细解读 k 的符号。
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