Families of Solutions, General Solutions and Particular Solutions | 解族、通解与特解

📚 Families of Solutions, General Solutions and Particular Solutions | 解族、通解与特解

A differential equation such as dy/dx = 2x has infinitely many solutions. Each value of the constant of integration gives a different curve, so the whole set is called a family of solutions.

像 dy/dx = 2x 这样的微分方程有无穷多个解。积分常数每取一个值就得到一条不同的曲线，因此整个集合称为解族。

In AQA A-Level Mathematics, you need to distinguish between the general solution, which contains an arbitrary constant, and a particular solution, which is selected from that family using an initial condition.

在 AQA A-Level 数学中，你需要区分含有任意常数的通解，以及利用初始条件从解族中选出的特解。

1. Differential Equations and Their Solutions | 微分方程及其解

A differential equation links a function y with its derivative, such as dy/dx = 2x. A solution is any function y = f(x) that makes the equation true when substituted.

微分方程将函数 y 与其导数联系起来，例如 dy/dx = 2x。解是代入后使方程成立的任何函数 y = f(x)。

For example, y = x² + 1 is a solution of dy/dx = 2x because differentiating gives y’ = 2x, exactly matching the right-hand side.

例如，y = x² + 1 是 dy/dx = 2x 的一个解，因为求导后得到 y’ = 2x，恰好等于方程右边。

However, y = x² + 5, y = x² − 7 and y = x² + 100 also work. This already shows that a differential equation usually has more than one solution.

然而，y = x² + 5、y = x² − 7 和 y = x² + 100 也同样成立。这已经说明微分方程通常有多个解。

2. Why a Constant Appears | 为什么会出现常数

When you integrate a derivative to find y, the indefinite integral always introduces an arbitrary constant, usually written C. This constant can be any real number, so it generates infinitely many functions.

当你对导数积分求 y 时，不定积分总会引入一个任意常数，通常记作 C。这个常数可以是任意实数，因此会产生无穷多个函数。

∫ 2x dx = x² + C

The collection y = x² + C is a family of parabolas. They have the same shape and only differ by a vertical translation.

集合 y = x² + C 是一族抛物线。它们形状相同，只是沿竖直方向平移。

3. General Solution of a First-Order Equation | 一阶方程的通解

The general solution of a first-order differential equation is an expression that contains one arbitrary constant and represents every possible solution. For dy/dx = 2y, the general solution is y = Ae²ˣ, where A is arbitrary.

一阶微分方程的通解是含有一个任意常数的表达式，它能表示所有可能的解。对于 dy/dx = 2y，通解为 y = Ae²ˣ，其中 A 为任意常数。

dy/dx = 2y ⇒ y = Ae²ˣ

If an equation is second-order, two constants appear. For example, solving d²y/dx² = 6x gives y = x³ + Ax + B after two integrations.

如果方程是二阶的，则会出现两个常数。例如，求解 d²y/dx² = 6x 得到 y = x³ + Ax + B，因为经历了两次积分。

4. Particular Solution and Initial Conditions | 特解与初始条件

A particular solution is obtained from the general solution by choosing a specific value of the arbitrary constant. This is usually done using an initial condition such as y(0) = 3.

特解是通过给任意常数选定一个具体值而从通解中得到的。这通常通过初始条件来完成，例如 y(0) = 3。

For dy/dx = 2y, substituting x = 0, y = 3 into y = Ae²ˣ gives 3 = Ae⁰, so A = 3. The particular solution is y = 3e²ˣ.

对于 dy/dx = 2y，将 x = 0、y = 3 代入 y = Ae²ˣ 得到 3 = Ae⁰，因此 A = 3。特解为 y = 3e²ˣ。

An initial condition is a pair of values, usually written as y(x₀) = y₀, that selects exactly one curve from the whole family.

初始条件是一对数值，通常写成 y(x₀) = y₀，它从整个解族中恰好选出一条曲线。

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