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单词 DifferentialEquation
释义

differential equation


A differential equationMathworldPlanetmath is an equation involving an unknown functionof one or more variables, its derivativesPlanetmathPlanetmath and the variables.This type of equations comes up often in many different branches ofmathematics. They are also especially important in many problems inphysics and engineering.

There are many types of differential equations. An ordinary differentialequation (ODE) is a differential equation where the unknown function dependson a single variable. A general ODE has the form

F(x,f(x),f(x),,f(n)(x))=0,(1)

where the unknown f is usually understood to be a real or complex valued function of x, and x is usually understood to be either a real or complexvariable.The of a differential equation is the order of the highest derivative appearingin Eq. (1). In this case, assuming that F dependsnontrivially on f(n)(x), the equation is of nth order.

If a differential equation is satisfied by a function whichidentically vanishes (i.e. f(x)=0 for each x in the of interest),then the equation is said to be homogeneousPlanetmathPlanetmath. Otherwise it is said to benonhomogeneous (or inhomogeneous). Many differential equationscan be expressed in the form

L[f]=g(x),

where L is a differentialoperatorMathworldPlanetmath (with g(x)=0 for the homogeneous case). If the operatorMathworldPlanetmath L islinear in f, then the equation is said to be a linear ODE andotherwise nonlinear.

Other types of differential equations involve more complicated involving the unknown function.A partial differential equation (PDE) is a differential equationwhere the unknown function depends on more than one variable. In a delay differential equation (DDE), the unknown function depends on the state of the system at some instant in the past.

Solving differential equations is a difficult task. Three major types of approaches are possible:

  • Exact methods are generally to equations of low order and/or to linear systems.

  • Qualitative methods do not give explicit for the solutions, but provide pertaining to the asymptotic behavior of the system.

  • Finally, numerical methods allow to construct approximated solutions.

Examples

A common example of an ODE is the equation for simple harmonic motion

d2udx2+ku=0.

This equation is of second orderPlanetmathPlanetmath. It can be transformed into a system of two first order differential equations by introducing a variable v=du/dx. Indeed, we then have

dvdx=-ku
dudx=v.

A common example of a PDE is the wave equationMathworldPlanetmath in three dimensions

2ux2+2uy2+2uz2=c22ut2
Titledifferential equation
Canonical nameDifferentialEquation
Date of creation2013-03-22 12:41:22
Last modified on2013-03-22 12:41:22
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id11
Authorrspuzio (6075)
Entry typeTopic
Classificationmsc 35-00
Classificationmsc 34-00
Related topicHeatEquation
Related topicMethodOfUndeterminedCoefficients
Related topicExampleOfUniversalStructure
Related topicCauchyInitialValueProblem
Related topicEquation
Related topicMaxwellsEquations
Definesordinary differential equation
DefinesODE
Definespartial differential equation
DefinesPDE
Defineshomogeneous
Definesnonhomogeneous
Definesinhomogeneous
Defineslinear differential equation
Definesnonlinear differential equation
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更新时间:2025/5/4 18:48:30