微分方程定义

微分方程定义

概念: 微分方程、常微分方程、常微分方程、偏微分方程、阶、解、通解、特解、奇解、定解条件、初值条件

微分方程：凡含有未知函数的导数或微分的方程

常微分方程：未知函数为一元函数的微分方程.

偏微分方程：未知函数为多元函数，同时含有多元函数的偏导数的微分方程

阶:微分方程中,未知函数的最高阶导数(或微分)的阶数

解:如果某个函数代入微分方程后使其两端恒等,则称此函数为该方程的解

通解(general solution):如果微分方程的解所包含独立的任意常数的个数等于方程的阶数,则称此解为方程的通解.（通解并不一定包含方程所有的解）

特解(particular solution)：微分方程任一确定的解

奇解：不包含在通解中的解

定解条件：用来确定微分方程特解的条件。（微分方程一般具有无数个解，为了确定微分方程的一个特解，必须给出这个解所满足的条件。）

初值条件：如果定解条件是由系统在初始时刻所处的状态给出，则也称这种定解条件

为初值条件。

In general,a differential equation is an equation that contains an unknown function and one or more of its derivatives.

The order of a differential equation is the order of the highest derivative that occurs in the equation.

A function f is called a solution of a differential equation if the equation is satisfied when y=f(x) and its derivatives are substituded into the equation.

An nth-order equation has an nth-parameter family of solution.

A separable equation is a first-order differential equation in which the expression for dy/dx can be factored as a function of x times a funtion of y. In other words,it can be written in the form dy/dx=g(x)f(y)

Homogeneous Equations:The first-order differential equation is homogeous if it can be put in the form y′=f(y/x)(x≠0)

A first-order linear differential equation is one that can be put into the form dy/dx+P(x)y=Q(x)

To solve the linear differential equation y′+P(x)y=Q(x),multiply both sides by the intergrating factor I(x)=e

p(x)dx and integrate both sides.