Differential Equations of the Form dy/dx = f(x)g(y) | 形如 dy/dx = f(x)g(y) 的微分方程

📚 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 AQA A-Level Mathematics, first-order differential equations often appear in the form dy/dx = f(x)g(y), where the right-hand side is a product of a function of x and a function of y. This structure allows us to separate the variables and integrate both sides, turning a differential equation into an algebraic problem. Mastering this technique is essential for the pure mathematics papers and for modelling real-world rates of change.

在 AQA A-Level 数学中，一阶微分方程常以 dy/dx = f(x)g(y) 的形式出现，其中右边是 x 的函数与 y 的函数的乘积。这种结构允许我们分离变量并对两边积分，从而把微分方程转化为代数问题。掌握这一技巧对纯数学试卷以及现实变化率建模都至关重要。

1. Recognising Separable First-Order Differential Equations | 识别可分离一阶微分方程

A first-order differential equation is separable if it can be written as dy/dx = f(x)g(y). In this form, one factor depends only on x and the other factor depends only on y. For example, dy/dx = x² sin y has f(x) = x² and g(y) = sin y, while dy/dx = 6x(y² + 1) has f(x) = 6x and g(y) = y² + 1.

一阶微分方程如果可以写成 dy/dx = f(x)g(y) 的形式，就称为可分离变量的微分方程。在这种形式中，一个因式只依赖于 x，另一个因式只依赖于 y。例如，dy/dx = x² sin y 中 f(x) = x²，g(y) = sin y；而 dy/dx = 6x(y² + 1) 中 f(x) = 6x，g(y) = y² + 1。

Some equations need to be rearranged before the separable form is clear. The equation dy/dx = y/x can be rewritten as dy/dx = y × (1/x), so f(x) = 1/x and g(y) = y. Similarly, dy/dx = x + xy can be factorised to dy/dx = x(1 + y), giving f(x) = x and g(y) = 1 + y.

有些方程需要先整理才能看出可分离形式。方程 dy/dx = y/x 可以改写为 dy/dx = y × (1/x)，因此 f(x) = 1/x，g(y) = y。类似地，dy/dx = x + xy 可以因式分解为 dy/dx = x(1 + y)，得到 f(x) = x 和 g(y) = 1 + y。

2. The Separation of Variables Method | 分离变量法

The separation method relies on treating dy/dx as a ratio of differentials, which is justified for separable first-order equations. We divide both sides by g(y) and multiply by dx to obtain 1/g(y) dy = f(x) dx. Then we integrate both sides with respect to their own variables.

分离变量法依赖于将 dy/dx 视为微分的比值，这对可分离的一阶微分方程是成立的。我们将两边除以 g(y) 并乘以 dx，得到 1/g(y) dy = f(x) dx。然后我们对自己的变量对两边进行积分。

The general algorithm is: first, separate the variables so that all y terms are on one side and all x terms are on the other. Second, integrate both sides, adding one arbitrary constant of integration to the right-hand side. Third, simplify and rearrange to obtain the general solution, usually involving one arbitrary constant.

一般步骤是：首先分离变量，使所有 y 项在一边，所有 x 项在另一边。其次，对两边积分，并在右边添加一个任意积分常数。第三，化简并整理得到通解，通常含有一个任意常数。

∫ 1/g(y) dy = ∫ f(x) dx

It is important to add only one arbitrary constant, not two. Since the sum of two constants is still a constant, a single constant C on one side is sufficient and keeps the solution simpler.

重要的是只添加一个任意常数，而不是两个。因为两个常数之和仍然是常数，所以一边的一个常数 C 就足够了，并且可以使解更简洁。

3. Step-by-Step Worked Example: dy/dx = 3x² y | 分步示例：dy/dx = 3x² y

Solve the differential equation dy/dx = 3x² y. First, separate the variables by dividing both sides by y and multiplying by dx. This gives 1/y dy = 3x² dx. Integrating both sides gives ∫ 1/y dy = ∫ 3x² dx, so ln|y| = x³ + C.

解微分方程 dy/dx = 3x² y。首先，将两边除以 y 并乘以 dx，以分离变量。得到 1/y dy = 3x² dx。两边积分得到 ∫ 1/y dy = ∫ 3x² dx，因此 ln|y| = x³ + C。

ln|y| = x³ + C

Exponentiate both sides to remove the logarithm: |y| = e^{x³ + C} = e^C e^x³. Since e^C is a positive constant, write A = ±e^C, which can be any non-zero real constant. The general solution is therefore y = A e^x³.

对两边取指数以去掉对数：|y| = e^{x³ + C} = e^C e^x³。由于 e^C 是正常数，令 A = ±e^C，A 可以是任何非零实数。因此通解为 y = A e^x³。

The absolute value in ln|y| can be removed because A accounts for the sign of y. If the original equation allows y = 0, then A = 0 may also be included, but this must be checked separately.

ln|y| 中的绝对值可以去掉，因为 A 已经考虑了 y 的符号。如果原方程允许 y = 0，那么 A = 0 也可能包含在内，但这需要单独检验。

4. Finding General Solutions with an Arbitrary Constant | 求含任意常数的通解

When an indefinite integral produces an arbitrary constant, it is often convenient to rename it to absorb factors, signs, or exponentials. For instance, C in ln|y| = x³ + C becomes A after exponentiation, and A can be positive, negative, or zero if the zero solution is included.

当不定积分产生任意常数时，通常可以方便地对其重新命名，以吸收因子、符号或指数。例如，ln|y| = x³ + C 中的 C 在取指数后变为 A，A 可以是正数、负数，或者如果包含零解，也可以为零。

The arbitrary constant must not be forgotten. A solution without a constant usually represents only one member of the family of curves. The general solution of a first-order differential equation should always contain one arbitrary constant.

任意常数不能忘记。没有常数的解通常只表示曲线族中的一个成员。一阶微分方程的通解应始终含有一个任意常数。

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