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

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

When you integrate a derivative such as dy/dx = 2x, you do not obtain a single function. You obtain y = x² + c, where c can be any real number. Each choice of c gives one possible curve. The whole collection is called a family of solutions. This article explains how AQA A-Level questions ask you to move from a family to the general solution, and then to a particular solution using given conditions.

当你对 dy/dx = 2x 这样的导数进行积分时，得到的并不是一个函数，而是 y = x² + c，其中 c 可以是任意实数。每一个 c 值对应一条可能的曲线。整个集合称为解族。本文将解释 AQA A-Level 题目如何要求你从解族写出通解，再利用给定条件求出特解。

1. The Idea of a Solution Family | 解族的概念

A differential equation such as dy/dx = f(x) or dy/dx = f(x, y) usually has infinitely many solutions. Each solution is obtained by giving the arbitrary constant a specific value. The set of all possible solutions is called the family of solutions.

像 dy/dx = f(x) 或 dy/dx = f(x, y) 这样的微分方程通常有无穷多个解。给任意常数一个具体值，就会得到其中一个解。所有可能解构成的集合称为解族。

For example, the equation dy/dx = 2x has solutions y = x² + c. For c = 0, y = x². For c = 1, y = x² + 1. These curves are different members of the same family.

例如，方程 dy/dx = 2x 的解为 y = x² + c。当 c = 0 时，y = x²；当 c = 1 时，y = x² + 1。这些曲线都是同一解族中的不同成员。

2. The Arbitrary Constant | 任意常数

Every time you perform an indefinite integral to solve a first-order differential equation, one arbitrary constant appears. If no extra information is given, the answer must include this constant.

每当你做一次不定积分来求解一阶微分方程，就会出现一个任意常数。如果没有额外信息，答案中必须包含这个常数。

∫ 2x dx = x² + c

Leaving out + c is not a small slip; it changes an infinite family of solutions into just one function.

漏掉

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）