高数线性代数的笔记：歪歪不挂高树，3

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今天小编为大家带来"微分方程程初识”，欢迎您的访问。

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Dear, this is Learning Yard.

Today Xiaobian brings you "Introduction to differential equations", you are welcome to visit.

开学了，不少大一的友友们也开始继续步入学习之旅了，今天歪歪就来为大家讲讲微分方程这一章节吧！

一.什么是微分方程

One. What are differential equations

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

二.微分方程的几种类型

Two. Several types of differential equations

1.常微分方程（Ordinary Differential Equation，ODE）指的是：仅含有一个独立变量的微分方程。如果微分方程中的未知函数包含两个或两个以上的独立变量，则该微分方程为偏微分方程。

2.可分离变量的微分方程

定义:形如dy/dx=f(x)g(y)的一阶微分方程,称为可分离变量的微分方程。如果方程能化为 ∫g(y)dy=∫f(x)dx，则就是分离变量的微分方程。

3.齐次微分方程（Homogeneous differential equation）是指能化为可分离变量方程的一类微分方程，它的标准形式是 y'=f(y/x)，其中 f 是已知的连续方程。

1.Ordinary Differential Equation (ODE) refers to a differential equation that contains only one independent variable. If an unknown function in a differential equation contains two or more independent variables, the differential equation is a partial differential equation.

2. Differential equations that separate variables

Definition: A first-order differential equation of the form dy/dx=f(x)g(y), called a differential equation of separable variables. If the equation can be reduced to ∫g(y)dy = ∫f(x)dx, it is a differential equation that separates the variables.

3. Homogeneous differential equations refer to a class of differential equations that can be reduced to separable variable equations, and its standard form is y'=f(y/x), where f is a known continuous equation.

三.微分方程的解决方法

Three. Solutions to differential equations

可分离变量的微分方程：

求解可分离变量的微分方程的方法为：将方程分离变量得到:g(y)dy=f(x)dx；等式两端求积分，得通解：∫g(y)dy=∫f(x)dx C。

形如f(x)g(y)dx=d(x)e(x)dy的方程叫做可分离变量微分方程。

齐次微分方程：

求解齐次微分方程的关键是作变换 u=y/x ，即 y=ux ，它可以把方程转换为关于 u 与 x 的可分离变量的方程，此时有 y'=u xu'，代入原方程即可得可分离变量的方程 u xu'=f(u) ,分离变量并积分即可得到结果，需要注意的是，最后应把 u=y/x 代入，并作必要的变形。

Differential equations for separable variables:

The method of solving the differential equation of separable variables is: separating the equation variables to get: g(y)dy=f(x)dx; Finding the integral at both ends of the equation yields a general solution: ∫g(y)dy=∫f(x)dx C.

An equation of the form f(x)g(y)dx=d(x)e(x)dy is called a separable variable differential equation.

Homogeneous differential equations:

The key to solving homogeneous differential equations is to transform u=y/x, that is, y=ux, which can convert the equation into an equation about the separable variables of u and x, at this time there is y'=u xu', substitute the original equation to obtain the equation u xu'=f(u) of the separable variable, separate the variable and integrate to obtain the result, it should be noted that finally u=y/x should be substituted, and the necessary deformation should be made.

四.重点考题类型——初值问题

Four. Key question types – preliminary value questions

对于初值问题的解法可分为通解和特解，其中只需要列出已知的表达式，求导最后进行验证是否漏解即可。最后给出两道例题，让大家理解一下初值问题的过程。

The solution to the initial value problem can be divided into general solution and special solution, in which only the known expressions need to be listed, and the derivation is finally carried out to verify whether the solution is missed. Finally, two example questions are given to let everyone understand the process of the initial value problem.

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参考资料：百度百科-秒懂百科、《高等数学辅导及习题精解》

翻译：有道翻译

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文字|歪歪

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