📚 Differential Equations of the Form dy/dx = f(x)g(y) | 形如 dy/dx = f(x)g(y) 的微分方程
In A-Level Mathematics, differential equations of the form dy/dx = f(x)g(y) are called separable differential equations. Their key feature is that the right-hand side can be written as a product of a function of x and a function of y. This structure allows us to separate variables and solve the equation by direct integration.
在 A-Level 数学中，形如 dy/dx = f(x)g(y) 的微分方程被称为可分离变量微分方程。其关键特征是等号右边可以写成 x 的函数与 y 的函数之乘积。这种结构使我们能够分离变量，并通过直接积分来求解方程。
1. Recognising the Form | 识别方程形式
The general form is dy/dx = f(x)g(y). To check if an equation is of this type, try to rewrite the right-hand side as a product of a term depending only on x and a term depending only on y. For example, dy/dx = x² sin(y) fits because f(x) = x² and g(y) = sin(y). Similarly, dy/dx = eˣ y is separable with f(x) = eˣ and g(y) = y.
一般形式为 dy/dx = f(x)g(y)。要判断一个方程是否属于这种类型，尝试将右边改写为一个仅依赖于 x 的项乘以一个仅依赖于 y 的项。例如，dy/dx = x² sin(y) 符合，因为 f(x) = x²，g(y) = sin(y)。类似地，dy/dx = eˣ y 是可分离的，其中 f(x) = eˣ，g(y) = y。
Equations that cannot be expressed in this product form, such as dy/dx = x + y, are not separable by this method and require other techniques such as integrating factors.
无法表示为这种乘积形式的方程，例如 dy/dx = x + y，不能用此方法分离变量，需要其他技巧如积分因子。
2. Separation of Variables: The Method | 分离变量法：方法步骤
The method involves four steps. First, rewrite the equation as (1/g(y)) dy = f(x) dx. Second, place integral signs on both sides: ∫ (1/g(y)) dy = ∫ f(x) dx. Third, integrate each side with respect to its own variable. Fourth, add the constant of integration and, if possible, solve for y in terms of x.
此方法包含四个步骤：首先，将方程改写为 (1/g(y)) dy = f(x) dx；其次，两边加积分号：∫ (1/g(y)) dy = ∫ f(x) dx；第三步，分别对各自的变量进行积分；最后，加上积分常数，并尽可能解出 y 关于 x 的表达式。
This procedure works because the variables are separated onto opposite sides of the equation, allowing independent integration.
这一过程之所以有效，是因为变量被分离到了方程的两边，从而可以独立积分。
3. Worked Example 1 | 例题解析 1
Solve the differential equation dy/dx = 3x² y.
求解微分方程 dy/dx = 3x² y。
First separate: (1/y) dy = 3x² dx. Integrate both sides: ∫ (1/y) dy = ∫ 3x² dx. This gives ln|y| = x³ + C. Exponentiate both sides to obtain y = Ae^(x³), where A = ±e^C is an arbitrary constant.
首先分离变量：(1/y) dy = 3x² dx。两边积分：∫ (1/y) dy = ∫ 3x² dx。得到 ln|y| = x³ + C。两边取指数，得到 y = Ae^(x³)，其中 A = ±e^C 为任意常数。
∫ (1/y) dy = ∫ 3x² dx → ln|y| = x³ + C → y = Ae^(x³)
Notice the absolute value inside the logarithm, which is essential for the domain of the solution.
注意对数内部的绝对值，这对解的定义域至关重要。
4. Worked Example 2 with Initial Conditions | 带初始条件的例题 2
Solve dy/dx = 2xy subject to y(0) = 3.
求解 dy/dx = 2xy，满足初值 y(0) = 3。
Separate variables: (1/y) dy = 2x dx. Integrate: ln|y| = x² + C. Thus y = Ae^(x²). Using y(0) = 3, we have 3 = Ae⁰ = A, so the particular solution is y = 3e^(x²).
分离变量：(1/y) dy = 2x dx。积分：ln|y| = x² + C。因此 y = Ae^(x²)。利用 y(0) = 3，得 3 = Ae⁰ = A，所以特解为 y = 3e^(x²)。
Initial conditions allow us to find the specific value of the arbitrary constant, turning a general solution into a particular solution.
初始条件允许我们确定任意常数的具体值，从而将通解转化为特解。
5. Special Case: dy/dx = f(x) | 特殊情况：dy/dx = f(x)
When g(y) is identically equal to 1, the equation reduces to dy/dx = f(x). This is a simple integration problem: y = ∫ f(x) dx + C. No separation step is needed because the equation is already separated.
当 g(y) 恒等于 1 时，方程退化为 dy/dx = f(x)。这是一个简单的积分问题：y = ∫ f(x) dx + C。不需要分离步骤，因为方程已经分离。
For example, dy/dx = 4x + 3 gives y = 2x² + 3x + C.
例如，dy/dx = 4x + 3 给出 y = 2x² + 3x + C。
6. Special Case: dy/dx = g(y) | 特殊情况：dy/dx = g(y)
When f(x) is a constant, say k, the equation becomes dy/dx = k g(y). Rearrange as (1/g(y)) dy = k dx and integrate: ∫ (1/g(y)) dy = kx + C.
当 f(x) 为常数 k 时，方程变为 dy/dx = k g(y)。整理为 (1/g(y)) dy = k dx 并积分：∫ (1/g(y)) dy = kx + C。
For instance, dy/dx = 0.2y models exponential growth. Separating gives ∫ (1/y) dy = ∫ 0.2 dx → ln|y| = 0.2x + C → y = Ae^(0.2x).
例如，dy/dx = 0.2y 模拟指数增长。分离变量得到 ∫ (1/y) dy = ∫ 0.2 dx → ln|y| = 0.2x + C → y = Ae^(0.2x)。
7. The Constant of Integration | 积分常数
The constant of integration C appears whenever we integrate. It represents an infinite family of curves. In many applications, an additional condition (like y(0) = 3) selects one unique solution.
每当我们积分时，积分常数 C 就会出现。它代表无穷多条曲线。在许多应用中，额外条件（如 y(0) = 3）会选择出唯一的解。
Sometimes C can be renamed or merged with other constants to simplify the expression. For example, e^(C) can be replaced by a new constant A, as long as we track sign changes.
有时 C 可以被重新命名或与其他常数合并以简化表达式。例如，e^(C) 可以用新常数 A 替代，只要注意符号变化。
8. Particular Solutions | 特解
A particular solution is obtained from the general solution by substituting the given initial condition and solving for the constant. Always check that the solution satisfies the original differential equation and the initial condition.
特解是通过将给定的初始条件代入通解并求出常数而得到的。务必验证解满足原微分方程和初始条件。
For example, for dy/dx = ky with y(0) = y₀, the particular solution is y = y₀ e^(kx), where y₀ is the initial value. This solution is unique for a given y₀.
例如，对于 dy/dx = ky，y(0) = y₀，特解为 y = y₀ e^(kx)，其中 y₀ 是初始值。在给定 y₀ 时，该解是唯一的。
9. Real-World Applications | 实际应用
Separable differential equations appear in many real-world contexts. Population growth, radioactive decay, Newton’s law of cooling, and the spread of disease are classic examples. In each case, the rate of change of a quantity is proportional to the quantity itself, leading to an equation of the form dy/dx = k y.
可分离变量微分方程出现在许多实际场景中。人口增长、放射性衰变、牛顿冷却定律以及疾病传播都是经典例子。在每种情况下，一个量的变化率与该量本身成正比，从而得到形式为 dy/dx = k y 的方程。
For Newton’s law of cooling, dT/dt = -k(T – T_env), where T is temperature and T_env is ambient temperature. Substitute u = T – T_env to get du/dt = -k u, which is separable.
对于牛顿冷却定律，dT/dt = -k(T – T_env)，其中 T 是温度，T_env 是环境温度。令 u = T – T_env，得到 du/dt = -k u，这是可分离的。
10. Common Mistakes | 常见错误
One common error is forgetting the constant of integration or misplacing it. Another is incorrectly separating variables, for example, writing ∫ g(y) dy = ∫ f(x) dx when the correct form is ∫ (1/g(y)) dy = ∫ f(x) dx. Also, when taking logarithms, remember that the absolute value is often needed.
一个常见错误是忘记积分常数或放错位置。另一个是错误地分离变量，例如写成 ∫ g(y) dy = ∫ f(x) dx，而正确形式应为 ∫ (1/g(y)) dy = ∫ f(x) dx。此外，当取对数时，经常需要考虑绝对值。
Students often fail to check whether the equation is truly separable. For instance, dy/dx = x + y cannot be written as f(x)g(y) because the sum does not factor in a simple way.
学生经常未能检查方程是否真正可分离。例如，dy/dx = x + y 不能写成 f(x)g(y)，因为和的形式无法简单因式分解。
11. Summary and Practice | 总结与练习
To summarise: identify the form dy/dx = f(x)g(y), separate the variables to (1/g(y)) dy = f(x) dx, integrate both sides, and apply initial conditions if given. Practice is essential to become comfortable with different algebraic manipulations.
总结：识别形如 dy/dx = f(x)g(y) 的方程，分离变量得到 (1/g(y)) dy = f(x) dx，两边积分，并在给出初始条件时应用它。练习对于熟练掌握不同的代数操作至关重要。
Try these problems: (1) dy/dx = x y²; (2) dy/dx = e^(x+y); (3) dy/dx = y cos(x) with y(0) = 2. Solutions involve separating variables and integrating.
尝试以下练习：(1) dy/dx = x y²；(2) dy/dx = e^(x+y)；(3) dy/dx = y cos(x) 且 y(0) = 2。解法涉及分离变量和积分。
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