Differential Equation - Wikipedia, The Free Encyclopedia

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Visualization of heat transfer in a pump casing, created bysolving the heat equation. Heat is being generated internally in

the casing and being cooled at the boundary, providing a

steady state temperature distribution.

Diffeenial eaionFrom Wikipedia, the free encyclopedia

A diffeenial eaion is a mathematical equation for an

unknown function of one or several variables that relates the

values of the function itself and its derivatives of various

orders. Differential equations play a prominent role in

engineering, physics, economics, and other disciplines.

Differential equations arise in many areas of science and

technology, specifically whenever a deterministic relation

involving some continuously varying quantities (modeled by

functions) and their rates of change in space and/or time

(expressed as derivatives) is known or postulated. This is

illustrated in classical mechanics, where the motion of a body

is described by its position and velocity as the time value

varies. Newton's laws allow one (given the position, velocity,

acceleration and various forces acting on the body) to express

these variables dynamically as a differential equation for the

unknown position of the body as a function of time. In somecases, this differential equation (called an equation of motion)

may be solved explicitly.

An example of modelling a real world problem using

differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air

resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air

resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This

means that the ball's acceleration, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a

function of time involves solving a differential equation.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas;

however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a

self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The

theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many

numerical methods have been developed to determine solutions with a given degree of accuracy.

Conen

1 Directions of study2 Nomenclature

2.1 Classification summary

2.1.1 Ordinary DE classification

2.2 Examples

3 Related concepts

4 Connection to difference equations

5 Universality of mathematical description

6 Exact solutions

7 Notable differential equations

7.1 Physics and engineering

7.2 Biology

7.3 Economics

8 See also

9 References

10 External links



Directions of stud

T , , , . A

. P

,

. D , ,

, , , . D

- , .. . I,

.

M ( ),

. T ,

, .

T .

omenclatureT

.

A (ODE) (

dependent variable) single . I ,

, , - -:

.

O order

. T first-order second-order differential equations. F , B'

( ) - . I

.

A (PDE) multiple . T

, , , ,

- , . S

mied

tpe.

B linear nonlinear. A linear

1 ( ) nonlinear . T

,

.Homogeneous

.. . T

() ;

constant coefficient linear differential equation.



There are very few methods of explicitly solving nonlinear differential equations; those that are known typically depend on the

equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended

time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions

for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard

problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier

Stokes existence and smoothness).

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid

under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum

equation that is valid for small amplitude oscillations (see below).

Classification summar

The mathematical definitions for the various classifications of differential equation can be collected as follows.

Ordinar DE classification

See main article: Ordinar differential equation.

In the table below,

All differential equations are of order n and arbitrary degree d .

F is an implicit function of: an independent variable , a dependent variable (a function of ), and integer derivatives of

(fractional derivatives are in fact possible, but not considered here).

may in general be a vector valued function:

so is an element of the vector space R , an element of a vector space of dimensionm, , where R is

the set of real numbers, is the cartesian product of R with itself m times to form an m-tuple

of real numbers.

This leads to a system of differential equations to be solved for 1, 2,... m.

is characterized by the function mapping .

r( ) is called a source term in , and A( ) is an arbitrary function, both assumed continuous in on defined intervals.

Characteristic Properties Differential equation

Implicit system of

dimension m

Explicit system of

dimension m

Autonomous: No

dependence

Linear: nth derivative



ca be ie

a a linear

combination f he

he

deiaie

Homogeneous:Sce e

i e

Nice he aig f ced he a f x, , ad he n (n-1) deiaie f

he i, i geea iici.

Eamples

I he fi g f eae, e u be a fci f x, ad c ad ae ca.

Ihgee fi-de iea ca cefficie dia diffeeia eai:

Hgee ecd-de iea dia diffeeia eai:

Hgee ecd-de iea ca cefficie dia diffeeia eai decibig he haic

cia:

Fi-de iea dia diffeeia eai:

Secd-de iea dia diffeeia eai decibig he i f a ed f egh L:

I he e g f eae, he fciu deed aiabe x ad t x ad y.

Hgee fi-de iea aia diffeeia eai:

Hgee ecd-de iea ca cefficie aia diffeeia eai f eiic e, he Laace

eai:



Third-order nonlinear partial differential equation, the Kortewegde Vries equation:

Related concepts

A delay differential equation (DDE) is an equation for a function of a single variable, usually calledtime, in which

the derivative of the function at a certain time is given in terms of the values of the function at earlier times.

A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and

the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion

equations.

A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in

implicit form.

Connection to difference equations

See also: Time scale calculus

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume

only discrete values, and the relationship involves values of the unknown function or functions and values at nearby

coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential

equations involve approximation of the solution of a differential equation by the solution of a corresponding difference

equation.

Universalit of mathematical description

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics,

differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations

first developed together with the sciences where the equations had originated and where the results found application.

However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential

equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind

diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface

of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which

allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory

of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat

equation. It turned out that many diffusion processes, while seemingly different, are described by the same equation; Black

Scholes equation in finance is for instance, related to the heat equation.

Eact solutions

Some differential equations have solutions which can be written in an exact and closed form. Several important classes are

given here.

In the table below, H ( ), Z ( ), H ( ), Z ( ), or H ( , ), Z ( , ) are any integrable functions of or (or both), and A, B, C , I ,

L, N , M are all constants. In general A, B, C , I , L, are real numbers, but N , M , P and Q may be complex. The differentialequations are in their equivalent and alternative forms which lead to the solution through integration.

Differential equation General solution



hee

olion ma be implici in and , obained b calclaing aboe inegal hen

ing

back-biing

If DE i eac o ha

hen he olion i:

hee and ae conan fncion fom he inegal ahe han

conan ale, hich ae

e o make he final fncion hold.

If DE i no eac, he fncion H ( , ) o Z ( , ) ma be ed o obain an

inegaing faco, he DE i

hen mliplied hogh b ha faco and he olion poceed he ame a.

If

hen



8

If

then

If

then

9

where are the solutions of the polynomial of degree :

Note that 3 and 4 are special cases of 7, they are relatively common cases and included for completeness.

Similarly 8 is a special case of 9, but 8 is a relatively common form, particularly in simple physical and engineering problems.

otable differential equations

Phsics and engineering

Newton's Second Law in dynamics (mechanics)

Hamilton's equations in classical mechanics

Radioactive decay in nuclear physics

Newton's law of cooling in thermodynamics

The wave equationMaxwell's equations in electromagnetism

The heat equation in thermodynamics

Laplace's equation, which defines harmonic functions

Poisson's equation

Einstein's field equation in general relativity

The Schrödinger equation in quantum mechanics

The geodesic equation

The NavierStokes equations in fluid dynamics

The CauchyRiemann equations in complex analysis

The PoissonBoltzmann equation in molecular dynamics

The shallow water equations

Universal differential equation

The Lorenz equations whose solutions exhibit chaotic flow.



Biolog

Verhulst equation biological population growth

von Bertalanffy model biological individual growth

LotkaVolterra equations biological population dynamics

Replicator dynamics may be found in theoretical biology

Hodgkin-Huxley model - neural action potentials

Economics

The BlackScholes PDE

Exogenous growth model

Malthusian growth model

The Vidale-Wolfe advertising model

See also

Complex differential equation

Exact differential equationIntegral equations

Linear differential equation

PicardLindelöf theorem on existence and uniqueness of solutions

Numerical methods

References

D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.

A. D. Polyanin and V. F. Zaitsev, Handbook of Eact Solutions for Ordinar Differential Equations (2nd

edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.W. Johnson, A Treatise on Ordinar and Partial Differential Equations

(http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001) , John Wiley and Sons, 1913, in University

of Michigan Historical Math Collection (http://hti.umich.edu/u/umhistmath/)

E.L. Ince, Ordinar Differential Equations, Dover Publications, 1956

E.A. Coddington and N. Levinson, Theor of Ordinar Differential Equations, McGraw-Hill, 1955

P. Blanchard, R.L. Devaney, G.R. Hall, Differential Equations , Thompson, 2006

Calculus, Teach Yourself, P.Abbott and H. Neill, 2003 pages 266-277

Further Elementary Analysis, R.I.Porter, 1978, chapter XIX Differential Equations</ref>

Eternal linksLectures on Differential Equations (http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-

2010/video-lectures/) MIT Open CourseWare Videos

Online Notes / Differential Equations (http://tutorial.math.lamar.edu/classes/de/de.aspx) Paul Dawkins, Lamar

University

Differential Equations (http://www.sosmath.com/diffeq/diffeq.html) , S.O.S. Mathematics

Differential Equation Solver (http://publicliterature.org/tools/differential_equation_solver/) Java applet tool used to

solve differential equations.

Mathematical Assistant on Web (http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode) Symbolic

ODE tool, using Maxima

Exact Solutions of Ordinary Differential Equations (http://eqworld.ipmnet.ru/en/solutions/ode.htm)Collection of ODE and DAE models of physical systems (http://www.hedengren.net/research/models.htm)

MATLAB models

Notes on Diffy Qs: Differential Equations for Engineers (http://www.jirka.org/diffyqs/) An introductory textbook on

differential equations by Jiri Lebl of UIUC



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Differential Equation - Wikipedia, The Free Encyclopedia