📚 Differential Equations of the Form dy/dx = f(x)g(y) | 形式为 dy/dx = f(x)g(y) 的微分方程
In AQA A-Level Mathematics, solving differential equations of the form dy/dx = f(x)g(y) is a core skill. This type is called separable because the variables can be separated onto opposite sides of the equation before integration. The chapter brings together differentiation, integration, logarithms and exponentials, and it is frequently examined in both pure and modelling contexts.
在 AQA A-Level 数学中，求解形如 dy/dx = f(x)g(y) 的微分方程是一项核心技能。这类方程被称为可分离变量方程，因为变量可以在积分前被分离到方程的两侧。本章综合了求导、积分、对数与指数函数，并且在纯数学和建模题中经常考查。
1. What Makes a Differential Equation Separable? | 什么是可分离变量的微分方程？
A differential equation is separable if it can be rearranged into the form dy/dx = f(x)g(y), where f(x) depends only on x and g(y) depends only on y. This structure allows us to move all y terms to the left and all x terms to the right before integrating.
如果一个微分方程可以整理为 dy/dx = f(x)g(y) 的形式，其中 f(x) 只依赖 x，g(y) 只依赖 y，那么它就是可分离变量的。这种结构允许我们在积分前将所有含 y 的项移到左边，将所有含 x 的项移到右边。
For example, dy/dx = x²y, dy/dx = eˣcos y and dy/dx = y ln x are all separable. However, dy/dx = x + y is not separable because the right-hand side cannot be written as a product of a function of x and a function of y.
例如，dy/dx = x²y、dy/dx = eˣcos y 和 dy/dx = y ln x 都是可分离变量方程。然而 dy/dx = x + y 不是可分离变量的，因为右侧无法写成只含 x 的函数与只含 y 的函数的乘积。
2. The Method of Separation of Variables | 分离变量法
To solve a separable equation, first divide both sides by g(y), provided g(y) is not zero, and multiply both sides by dx. This produces the separated form shown below.
求解可分离变量方程时，首先在 g(y) 不为零的前提下将两边除以 g(y)，再将两边乘以 dx。这样可得到如下分离形式。
∫ (1/g(y)) dy = ∫ f(x) dx
Always write one constant of integration, usually on the right-hand side. This constant represents the whole family of antiderivatives and must not be forgotten.
始终只写一个积分常数，通常写在右侧。这个常数代表整个原函数族，绝不能遗漏。
Some equations may need a little algebra first. For instance, dy/dx = (x + y)/x can be rearranged to dy/dx = 1 + y/x, which is not separable, but the original form can sometimes be separated after simplification if it is a product.
有些方程可能需要先进行代数整理。例如 dy/dx = (x + y)/x 可以整理为 dy/dx = 1 + y/x，这并不是可分离变量形式；但如果原式能化为乘积形式，有时可在化简后分离。
3. Integrating Both Sides | 两边积分
After separating variables, integrate each side with respect to its own variable. The constant of integration can be added on either side, but by convention it is placed on the x side.
分离变量后，对每一侧关于各自的变量进行积分。积分常数可以加在任意一侧，但习惯上放在含 x 的一侧。
For instance, if dy/dx = 2xy, then (1/y) dy = 2x dx. Integrating gives ln |y| = x² + C.
例如，若 dy/dx = 2xy，则 (1/y) dy = 2x dx。积分得 ln |y| = x² + C。
ln |y| = x² + C
When logarithms appear, you can often remove them by exponentiating both sides to obtain an explicit solution. However, be careful with absolute value signs when the solution can be negative.
当出现对数时，通常可以对两边取指数来消去对数，从而得到显式解。但当解可能为负时，要小心处理绝对值符号。
4. Finding General Solutions | 求通解
The general solution of a first-order separable equation usually contains one arbitrary constant. It can be left in implicit form, such as y² = x² + C, or rearranged into explicit form y = ±√(x² + C) if possible.
一阶可分离变量方程的通解通常含有一个任意常数。它可以保留为隐式形式，例如 y² = x² + C；如果可能，也可以整理为显式形式 y = ±√(x² + C)。
AQA often accepts any correct equivalent form, but you should simplify as much as possible unless the question states otherwise. If the question asks for the solution in the form y = f(x), you must give an explicit expression.
AQA 考试通常接受任何正确的等价形式，但除非题目另有说明，你应尽量化简。如果题目要求写成 y = f(x) 的形式，则必须给出显式表达式。
For example, from dy/dx = x/y, separation gives y dy = x dx, so ½ y² = ½ x² + C. Multiplying by 2 gives y² = x² + 2C, and since 2C is just another constant, we can write y² = x² + C.
例如，由 dy/dx = x/y，分离得 y dy = x dx，所以 ½ y² = ½ x² + C。两边乘以 2 得 y² = x² + 2C；由于 2C 仍然是任意常数，因此可写为 y² = x² + C。
5. Applying Initial Conditions | 应用初始条件
To find a particular solution, substitute the given initial condition, such as y(x₀) = y₀, into the general solution and solve for C. This is often the final step in an exam question.
要求特解，需要将给定的初始条件（例如 y(x₀) = y₀）代入通解，并解出 C 的值。这通常是考试题的最后一步。
Example: solve dy/dx = y with y(0) = 3. Separating gives (1/y) dy = dx, so ln |y| = x + C. Substituting x = 0 and y = 3 gives C = ln 3, hence y = 3eˣ.
例题：求解 dy/dx = y，且 y(0) = 3。分离变量得 (1/y) dy = dx，所以 ln |y| = x + C。代入 x = 0，y = 3，得 C = ln 3，因此 y = 3eˣ。
y = 3eˣ
If the constant comes from a logarithm, combine it using log rules before exponentiating. This keeps the working neat and reduces the risk of sign errors.
如果常数来自对数，在取指数前先用对数法则进行合并。这样可以使过程更整洁，并减少符号错误的风险。
6. Verifying a Solution | 验证解
You can verify a solution by differentiating it and checking that it satisfies the original differential equation. If the solution is implicit, use implicit differentiation.
你可以通过对解求导并检查它是否满足原微分方程来验证解。如果解是隐式的，则使用隐函数求导法。
A quick check can prevent sign or constant errors, especially when exponentiating logarithmic expressions. For example, if the solution is y² = x² + 3, differentiating gives 2y
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