Differential Equations

微分方程 Differential Equations, DE

基本概念和分类 Basic Concepts and Classification

微分方程的定义 Definition of Differential Equations

微分方程是一个包含未知函数及其导数的方程。

A differential equation is an equation that contains an unknown function and its derivatives.

数学表示：

Mathematical representation:

$f(x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)})=0$

其中 $y=y(x)$ 是未知函数，$y^{\prime },y^{\prime \prime },\dots ,y^{(n)}$ 是 $y$ 的各阶导数。

Where $y=y(x)$ is the unknown function, and $y^{\prime },y^{\prime \prime },\dots ,y^{(n)}$ are the derivatives of $y$ of various orders.

重要性：

Importance:

1. 在数学中建立函数关系 Establish functional relationships in mathematics
2. 描述物理、工程、经济等领域的动态系统 Describe dynamic systems in physics, engineering, economics, etc.
3. 预测和模拟复杂系统的行为 Predict and simulate the behavior of complex systems

微分方程的分类 Classification of Differential Equations

1. 按未知函数的自变量数目分类： Classification based on the number of independent variables of the unknown function:

   a) 常微分方程 (Ordinary Differential Equations, ODE)：

   只包含一个自变量的导数

   Contains derivatives with respect to only one independent variable

   例如 (Example): $\frac{dy}{dx}+2y=x$

   b) 偏微分方程 (Partial Differential Equations, PDE)：

   包含多个自变量的偏导数

   Contains partial derivatives with respect to multiple independent variables

   例如 (Example): $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$

2. 按最高阶导数分类： Classification based on the highest order of derivatives:

   a) 一阶微分方程 (First-order Differential Equations)：

   最高阶导数为一阶

   The highest order derivative is first-order

   例如 (Example): $\frac{dy}{dx}=x^2+y$

   b) 二阶微分方程 (Second-order Differential Equations)：

   最高阶导数为二阶

   The highest order derivative is second-order

   例如 (Example): $\frac{d^{2}y}{d{x}^2}+3\frac{dy}{dx}+2y=0$

   c) 高阶微分方程 (Higher-order Differential Equations)：

   最高阶导数大于二阶

   The highest order derivative is higher than second-order

   例如 (Example): $\frac{d^{3}y}{d{x}^3}+2\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}+y=\sin x$

3. 按方程的形式分类： Classification based on the form of the equation:

   a) 线性微分方程 (Linear Differential Equations)：

   未知函数及其导数均以一次方出现，且系数为自变量的函数或常数

   The unknown function and its derivatives appear only to the first power, and their coefficients are functions of the independent variable or constants

   例如 (Example): $\frac{d^{2}y}{d{x}^2}+p(x)\frac{dy}{dx}+q(x)y=r(x)$

   b) 非线性微分方程 (Nonlinear Differential Equations)：

   不满足线性方程条件的方程

   Equations that do not satisfy the conditions for linear equations

   例如 (Example): $\frac{dy}{dx}=y^2+\sin x$