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第232行：
第232行： − + 第239行：
第239行： − 在第一组例子中，u 是 x 和 c 的未知函数，它们是应该已知的常数。常微分方程和偏微分方程的两种广义分类包括区分线性和非线性微分方程，以及区分齐次微分方程和非齐次微分方程。+ − + − <math> \frac{du}{dx} = cu+x^2. </math> − − 数学[ frac }{ dx } cu + x ^ 2。数学 − + − − − <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math> − − 数学框架 ^ 2u }{ dx ^ 2}-框架{ dx } + u 0。数学 − − + − − − <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math> − − 数学框架 ^ 2u }{ dx ^ 2} + omega ^ 2u 0。数学 − − + − − − <math> \frac{du}{dx} = u^2 + 4. </math> − − 2 + 4.数学 − + − − <math> L\frac{d^2u}{dx^2} + g\sin u = 0. </math> − − 数学 l frac { d ^ 2 u }{ dx ^ 2} + g sin u 0。数学 第307行：
第283行： − + − + − <math> \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. </math> − − 数学部分 u } + t frac 部分 u }0。数学 − + − <math> \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. </math> − − 部分 x ^ 2} + 部分 y ^ 2}0。数学 − + − − <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math> − − 数学 frac { partial u }6u frac { partial u }-frac { partial ^ 3 u }。数学
→Types
===Examples===
===Examples===
示例
In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.
In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.
在第一组例子中，待求解的''u''是''x''的函数，''c''和''ω''是应该已知的常数。常微分方程和偏微分方程的两种广义分类要区分微分方程的线性和非线性，以及区分微分方程的齐次和非齐次。
* Heterogeneous first-order linear constant coefficient ordinary differential equation:
* Heterogeneous first-order linear constant coefficient ordinary differential equation:
非齐次一阶常系数常微分方程
:: <math> \frac{du}{dx} = cu+x^2. </math>
:: <math> \frac{du}{dx} = cu+x^2. </math>
* Homogeneous second-order linear ordinary differential equation:
* Homogeneous second-order linear ordinary differential equation:
齐次二阶线性常微分方程
:: <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>
:: <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>
* Homogeneous second-order linear constant coefficient ordinary differential equation describing the [[harmonic oscillator]]:
* Homogeneous second-order linear constant coefficient ordinary differential equation describing the [[harmonic oscillator]]:
用于描述简谐振动的齐次二阶常系数常系数微分方程
:: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>
:: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>
* Heterogeneous first-order nonlinear ordinary differential equation:
* Heterogeneous first-order nonlinear ordinary differential equation:
非齐次一阶非线性常微分方程
:: <math> \frac{du}{dx} = u^2 + 4. </math>
:: <math> \frac{du}{dx} = u^2 + 4. </math>
* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a [[pendulum]] of length ''L'':
* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a [[pendulum]] of length ''L'':
用于描述摆长为L的钟摆运动的二阶非线性（因正弦函数产生）常微分方程
:: <math> L\frac{d^2u}{dx^2} + g\sin u = 0. </math>
:: <math> L\frac{d^2u}{dx^2} + g\sin u = 0. </math>
In the next group of examples, the unknown function u depends on two variables x and t or x and y.
In the next group of examples, the unknown function u depends on two variables x and t or x and y.
在下一组例子中，未知函数 u 依赖于两个变量 x 和 t 或 x 和 y。
在下一组例子中，未知函数''u''依赖于两个变量''x'' 和 ''t''或者''x''和''y''。
* Homogeneous first-order linear partial differential equation:
* Homogeneous first-order linear partial differential equation:
齐次一阶线性偏微分方程
:: <math> \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. </math>
:: <math> \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. </math>
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
齐次二阶线性常系数椭圆偏微分方程，也称为拉普拉斯方程
:: <math> \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. </math>
:: <math> \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. </math>
* Homogeneous third-order non-linear partial differential equation :
* Homogeneous third-order non-linear partial differential equation :
齐次三阶非线性偏微分方程
:: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>
:: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>
==Existence of solutions==
==Existence of solutions==