Differential Equations and Solving Techniques | 微分方程与求解技巧

📚 Differential Equations and Solving Techniques | 微分方程与求解技巧

Differential equations are equations that involve unknown functions and their derivatives. They appear in physics, chemistry, economics, and engineering, making them one of the most powerful tools in applied mathematics.

微分方程是含有未知函数及其导数的方程。它们广泛出现在物理、化学、经济和工程中，是应用数学中最强大的工具之一。

1. What is a Differential Equation? | 什么是微分方程

A differential equation relates a function y(t) with its derivatives, such as dy/dt or d²y/dt². For example, dy/dx = 3x² is a simple first-order differential equation.

微分方程把函数 y(t) 与其导数联系起来，例如 dy/dt 或 d²y/dt²。比如 dy/dx = 3x² 就是一个简单的一阶微分方程。

The solution of a differential equation is a function, not a single number. Solving often means finding y in terms of x.

微分方程的解是函数，而不是一个数字。求解通常意味着找到用 x 表示的 y。

2. Order, Degree and Linearity | 阶、次数与线性性

The order of a differential equation is the highest derivative present. A first-order equation involves dy/dx, while a second-order equation involves d²y/dx².

微分方程的阶是方程中出现的最高阶导数。一阶方程含有 dy/dx，二阶方程含有 d

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