diffrential equations wikipedia

8/2/2019 Diffrential Equations Wikipedia

1/9

Differential equation 1

Differential equation

Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat

is being generated internally in the casing and being cooled at the boundary, providing a

steady state temperature distribution.

A differential equation 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 byfunctions) 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 some cases, 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 determinedwithout 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.

Directions of study

The study of differential equations is a wide field in pure and applied mathematics, physics, meteorology, and

engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure

mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the

rigorous justification of the methods for approximating solutions. Differential equations play an important role in

modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to
http://en.wikipedia.org/w/index.php?title=Engineeringhttp://en.wikipedia.org/w/index.php?title=Meteorologyhttp://en.wikipedia.org/w/index.php?title=Physicshttp://en.wikipedia.org/w/index.php?title=Applied_mathematicshttp://en.wikipedia.org/w/index.php?title=Pure_mathematicshttp://en.wikipedia.org/w/index.php?title=Numerical_methodshttp://en.wikipedia.org/w/index.php?title=Dynamical_systemshttp://en.wikipedia.org/w/index.php?title=Equations_of_motionhttp://en.wikipedia.org/w/index.php?title=Newton%27s_laws_of_motionhttp://en.wikipedia.org/w/index.php?title=Classical_mechanicshttp://en.wikipedia.org/w/index.php?title=Classical_mechanicshttp://en.wikipedia.org/w/index.php?title=Deterministic_system_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Economicshttp://en.wikipedia.org/w/index.php?title=Physicshttp://en.wikipedia.org/w/index.php?title=Engineeringhttp://en.wikipedia.org/w/index.php?title=Derivativehttp://en.wikipedia.org/w/index.php?title=Variable_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Function_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Equationhttp://en.wikipedia.org/w/index.php?title=Mathematicshttp://en.wikipedia.org/w/index.php?title=File%3AElmer-pump-heatequation.pnghttp://en.wikipedia.org/w/index.php?title=Steady_statehttp://en.wikipedia.org/w/index.php?title=Heathttp://en.wikipedia.org/w/index.php?title=Heat_equation


2/9


interactions between neurons. Differential equations such as those used to solve real-life problems may not

necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using

numerical methods.

Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not

have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in

more physically reasonable properties of solutions, such as possible presence of shocks for equations of hyperbolictype.

The study of the stability of solutions of differential equations is known as stability theory.

Nomenclature

The theory of differential equations is quite developed and the methods used to study them vary significantly with

the type of the equation.

An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as

the dependent variable) is a function of a single independent variable. In the simplest form, the unknown

function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: thiscorresponds to considering a system of ordinary differential equations for a single function.

Ordinary differential equations are further classified according to the order of the highest derivative of the

dependent variable with respect to the independent variable appearing in the equation. The most important cases

for applications are first-order and second-order differential equations. For example, Bessel's differential

equation

(in which is the dependent variable) is a second-order differential equation. In the classical literature also

distinction is made between differential equations explicitly solved with respect to the highest derivative and

differential equations in an implicit form.

A partial differential equation (PDE) is a differential equation in which the unknown function is a function of

multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to

the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic

equations, especially for second-order linear equations, is of utmost importance. Some partial differential

equations do not fall into any of these categories over the whole domain of the independent variables and they are

said to be ofmixed type.

Both ordinary and partial differential equations are broadly classified as linear and nonlinear. A differential

equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and

nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace ofan appropriate function space, which results in much more developed theory of linear differential equations.

Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear

subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the

unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the

independent variable or variables; if these coefficients are constants then one speaks of a 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
http://en.wikipedia.org/w/index.php?title=Symmetrieshttp://en.wikipedia.org/w/index.php?title=Chaos_theoryhttp://en.wikipedia.org/w/index.php?title=Chaos_theoryhttp://en.wikipedia.org/w/index.php?title=Symmetrieshttp://en.wikipedia.org/w/index.php?title=Partial_derivativeshttp://en.wikipedia.org/w/index.php?title=Partial_differential_equationhttp://en.wikipedia.org/w/index.php?title=Bessel%27s_differential_equationhttp://en.wikipedia.org/w/index.php?title=Bessel%27s_differential_equationhttp://en.wikipedia.org/w/index.php?title=Matrix_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Vector-valued_functionhttp://en.wikipedia.org/w/index.php?title=Dependent_variable%23Use_in_mathematicshttp://en.wikipedia.org/w/index.php?title=Ordinary_differential_equationhttp://en.wikipedia.org/w/index.php?title=Stability_theoryhttp://en.wikipedia.org/w/index.php?title=Differentiablehttp://en.wikipedia.org/w/index.php?title=Weak_derivativehttp://en.wikipedia.org/w/index.php?title=Weak_solutionhttp://en.wikipedia.org/w/index.php?title=Numerical_ordinary_differential_equationshttp://en.wikipedia.org/w/index.php?title=Closed-form_expression


3/9


be a significant advance in the mathematical theory (cf. NavierStokes 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 summaryThe mathematical definitions for the various classifications of differential equation can be collected as follows.

Ordinary DE classification

See main article: Ordinary differential equation.

In the table below,

All differential equations are ofordern and arbitrary degreed.

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

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

y may in general be a vector valued function:

sox is an element of the vector space R, y an element of a vector space of dimension m, , where

R is the set of real numbers, is the cartesian product ofR with itselfm times to form

an m-tuple of real numbers.

This leads to a system of differential equations to be solved fory1,y

2,...y

m.

y is characterized by the function mapping .

r(x) is called a source term inx, andA(x) is an arbitrary function, both assumed continuous inx on defined intervals.

Characteristic Properties Differential equation

Implicit system of dimension m

Explicit system of dimension m

Autonomous: Nox dependence

Linear: nth derivative can be written

as a linear combination of the other

derivatives

Homogeneous: Source term is zero

Notice the mapping from or corresponds to the map fromx, y, and the n or (n-1) derivatives of

y to the solution, in general implicit.
http://en.wikipedia.org/w/index.php?title=Linear_combinationhttp://en.wikipedia.org/w/index.php?title=Map_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Cartesian_producthttp://en.wikipedia.org/w/index.php?title=Real_numbershttp://en.wikipedia.org/w/index.php?title=Set_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Vector_spacehttp://en.wikipedia.org/w/index.php?title=Element_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Vector_valued_functionhttp://en.wikipedia.org/w/index.php?title=Fractional_calculushttp://en.wikipedia.org/w/index.php?title=Implicit_functionhttp://en.wikipedia.org/w/index.php?title=Ordinary_differential_equationhttp://en.wikipedia.org/w/index.php?title=Linearizationhttp://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes_existence_and_smoothness


4/9


Examples

In the first group of examples, let u be an unknown function ofx, and c and are known constants.

Inhomogeneous first-order linear constant coefficient ordinary differential equation:

Homogeneous second-order linear ordinary differential equation:

Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic

oscillator:

Inhomogeneous first-order nonlinear ordinary differential equation:

Second-order nonlinear ordinary differential equation describing the motion of a pendulum of lengthL:

In the next group of examples, the unknown function u depends on two variablesx and torx andy.

Homogeneous first-order linear partial differential equation:

Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace

equation:

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 called time, in

which the derivative of the function at a certain time is given in terms of the values of the function at earliertimes.

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.
http://en.wikipedia.org/w/index.php?title=Differential_algebraic_equationhttp://en.wikipedia.org/w/index.php?title=Wiener_processhttp://en.wikipedia.org/w/index.php?title=Stochastic_processhttp://en.wikipedia.org/w/index.php?title=Stochastic_differential_equationhttp://en.wikipedia.org/w/index.php?title=Delay_differential_equationhttp://en.wikipedia.org/w/index.php?title=Korteweg%E2%80%93de_Vries_equationhttp://en.wikipedia.org/w/index.php?title=Laplace_equationhttp://en.wikipedia.org/w/index.php?title=Laplace_equationhttp://en.wikipedia.org/w/index.php?title=Pendulumhttp://en.wikipedia.org/w/index.php?title=Harmonic_oscillatorhttp://en.wikipedia.org/w/index.php?title=Harmonic_oscillator


5/9


Connection to difference equations

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.

Universality 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-orderpartial 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; BlackScholes equation in finance is for

instance, related to the heat equation.

Exact 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(x),Z(x),H(y),Z(y), orH(x,y),Z(x,y) are any integrable functions ofx ory (or both), andA,B,

C,I,L,N,Mare all constants. In general A,B, C,I,L, are real numbers, but N,M,P and Q may be complex. The

differential equations are in their equivalent and alternative forms which lead to the solution through integration.

Differential equation General solution

1

2

3

4

where

5

solution may be implicit inx andy, obtained by calculating above integral then using

back-substituting
http://en.wikipedia.org/w/index.php?title=Integralhttp://en.wikipedia.org/w/index.php?title=Integrablehttp://en.wikipedia.org/w/index.php?title=Black%E2%80%93Scholeshttp://en.wikipedia.org/w/index.php?title=Diffusionhttp://en.wikipedia.org/w/index.php?title=Heat_equationhttp://en.wikipedia.org/w/index.php?title=Joseph_Fourierhttp://en.wikipedia.org/w/index.php?title=Wave_equationhttp://en.wikipedia.org/w/index.php?title=Partial_differential_equationhttp://en.wikipedia.org/w/index.php?title=Mathematical_modellinghttp://en.wikipedia.org/w/index.php?title=Economicshttp://en.wikipedia.org/w/index.php?title=Biologyhttp://en.wikipedia.org/w/index.php?title=Chemistryhttp://en.wikipedia.org/w/index.php?title=Physicshttp://en.wikipedia.org/w/index.php?title=Difference_equations


6/9


6

7If DE is exact so that

then the solution is:

where and are constant functions from the integrals rather than constant values, which are

set to make the final function hold.

If DE is not exact, the functionsH(x,y) orZ(x,y) may be used to obtain an integrating factor, the DE is

then multiplied through by that factor and the solution proceeds the same way.

8 If

then

If

then

If

then

9

where are the dsolutions of the polynomial of degree d:

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.

Notable differential equations

Physics 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 equation Maxwell'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 Schrdinger equation in quantum mechanics

The geodesic equation

The NavierStokes equations in fluid dynamics

The CauchyRiemann equations in complex analysis

The Poisson

Boltzmann equation in molecular dynamics The shallow water equations
http://en.wikipedia.org/w/index.php?title=Shallow_water_equationshttp://en.wikipedia.org/w/index.php?title=Molecular_dynamicshttp://en.wikipedia.org/w/index.php?title=Poisson%E2%80%93Boltzmann_equationhttp://en.wikipedia.org/w/index.php?title=Complex_analysishttp://en.wikipedia.org/w/index.php?title=Cauchy%E2%80%93Riemann_equationshttp://en.wikipedia.org/w/index.php?title=Fluid_dynamicshttp://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/w/index.php?title=Geodesic%23%28pseudo-%29Riemannian_geometryhttp://en.wikipedia.org/w/index.php?title=Quantum_mechanicshttp://en.wikipedia.org/w/index.php?title=Schr%C3%B6dinger_equationhttp://en.wikipedia.org/w/index.php?title=General_relativityhttp://en.wikipedia.org/w/index.php?title=Einstein%27s_field_equationhttp://en.wikipedia.org/w/index.php?title=Poisson%27s_equationhttp://en.wikipedia.org/w/index.php?title=Harmonic_functionhttp://en.wikipedia.org/w/index.php?title=Laplace%27s_equationhttp://en.wikipedia.org/w/index.php?title=Thermodynamicshttp://en.wikipedia.org/w/index.php?title=Heat_equationhttp://en.wikipedia.org/w/index.php?title=Electromagnetismhttp://en.wikipedia.org/w/index.php?title=Maxwell%27s_equationshttp://en.wikipedia.org/w/index.php?title=Wave_equationhttp://en.wikipedia.org/w/index.php?title=Thermodynamicshttp://en.wikipedia.org/w/index.php?title=Newton%27s_law_of_coolinghttp://en.wikipedia.org/w/index.php?title=Nuclear_physicshttp://en.wikipedia.org/w/index.php?title=Radioactive_decayhttp://en.wikipedia.org/w/index.php?title=Hamilton%27s_equationshttp://en.wikipedia.org/w/index.php?title=Dynamics_%28mechanics%29http://en.wikipedia.org/w/index.php?title=Newton%27s_Second_Lawhttp://en.wikipedia.org/w/index.php?title=Degree_of_a_polynomialhttp://en.wikipedia.org/w/index.php?title=Polynomialhttp://en.wikipedia.org/w/index.php?title=Integrating_factor


7/9


8/9


References

[1] http://www.hti. umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001

[2] http://hti.umich. edu/u/umhistmath/

[3] http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/

[4] http://tutorial.math. lamar.edu/classes/de/de. aspx

[5] http://www.sosmath.com/diffeq/diffeq.html

[6] http://publicliterature.org/tools/differential_equation_solver/[7] http://user.mendelu. cz/marik/maw/index. php?lang=en&form=ode

[8] http://eqworld.ipmnet.ru/en/solutions/ode. htm

[9] http://www.hedengren. net/research/models. htm

[10] http://www.jirka.org/diffyqs/
http://www.jirka.org/diffyqs/http://www.hedengren.net/research/models.htmhttp://eqworld.ipmnet.ru/en/solutions/ode.htmhttp://user.mendelu.cz/marik/maw/index.php?lang=en&form=odehttp://publicliterature.org/tools/differential_equation_solver/http://www.sosmath.com/diffeq/diffeq.htmlhttp://tutorial.math.lamar.edu/classes/de/de.aspxhttp://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/http://hti.umich.edu/u/umhistmath/http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001


9/9

Article Sources and Contributors 9

Article Sources and ContributorsDifferential equation Source: http://en.wikipedia.org/w/index.php?oldid=477644758 Contributors: 17Drew, After Midnight, Ahoerstemeier, Alarius, Alfred Centauri, Amahoney, Andrei

Polyanin, Andres, AndrewHowse, Andycjp, Andytalk, AngryPhillip, Anonymous Dissident, Antoni Barau, Antonius Block, Anupam, Apmonitor, Arcfrk, Asdf39, Asyndeton, Attilios,

Babayagagypsies, [email protected], Baccyak4H, Bejohns6, Bento00, Berland, Bidabadi, Bigusbry, BillyPreset, Brandon, Bryanmcdonald, Btyner, Bygeorge2512, Callumds, Charles

Matthews, Chtito, Cispyre, Cmprince, ConMan, Cxz111, Cybercobra, DAJF, Danski14, Dbroadwell, Ddxc, Delaszk, DerHexer, Difu Wu, Djordjes, DominiqueNC, Donludwig, Dpv, Dr sarah

madden, Drmies, DroEsperanto, Dysprosia, EconoPhysicist, Elwikipedista, EricBright, Estudiarme, Fintor, Fioravante Patrone, Fioravante Patrone en, Friend of the Facts, Gabrielleitao,

Gandalf61, Gauss, Geni, Giftlite, Gombang, Grenavitar, Haham hanuka, Hamiltondaniel, Harry, Haruth, Haseeb Jamal, Heikki m, Ilya Voyager, Iquseruniv, Iulianu, Izodman2012, J arino,

J.delanoy, Ja 62, Jak86, Jao, Jauhienij, Jayden54, Jeancey, Jersey Devil, Jim Sukwutput, Jim.belk, Jim.henderson, JinJian, Jitse Niesen, JohnOwens, JorisvS, K-UNIT, Kayvan45622, KeithJonsn,Kensaii, Khalid Mahmood, Klaas van Aarsen, Kr5t, LOL, Lambiam, Lavateraguy, Lethe, LibLord, Linas, Lumos3, Madmath789, Mandarax, Mankarse, MarSch, Martynas Patasius, Maschen,

Math.geek3.1415926, Matqkks, Mattmnelson, Maurice Carbonaro, Maxis f tw, Mazi, McVities, Mduench, Mets501, Mh, Michael Hardy, Mindspillage, MisterSheik, Mohan1986, Mossaiby,

Mpatel, MrOllie, Mtness, Mysidia, Nik-renshaw, Nkayesmith, Oleg Alexandrov, Opelio, Pahio, Parusaro, Paul August, Paul Matthews, Paul Richter, Pgk, Phoebe, Pinethicket, PseudoSudo,

Qzd800, Rama's Arrow, Reallybored999, RexNL, Robin S, Romansanders, Rosasco, Ruakh, SDC, SFC9394, Salix alba, Sam Staton, Sampathsris, Senoreuchrestud, Silly rabbit, Siroxo, Skakkle,

Skypher, SmartPatrol, Snowjeep, Spirits in the Material, Starwiz, Suffusion of Yellow, Sverdrup, Symane, TVBZ28, TYelliot, Tannkrem, Tbsmith, TexasAndroid, Tgeairn, The Hybrid, The

Thing That Should Not Be, Timelesseyes, Tranum1234567890, Tuseroni, User A1, Vanished User 0001, Vthiru, Waldir, Waltpohl, Wavelength, Wclxlus, Wihenao, Willtron, Winterheart,

XJamRastafire, Yafujifide, Zepterfd, 301 anonymous edits

Image Sources, Licenses and ContributorsFile:Elmer-pump-heatequation.png Source: http://en.wikipedia.org/w/index.php?title=File:Elmer-pump-heatequation.pngLicense: Creative Commons Attribution-Sharealike 3.0

Contributors: Kri, User A1, 1 anonymous edits

License

Creative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/

diffrential equations wikipedia

Documents