Differential equation

Inmathematics,adifferential equationis anequationthat relates one or more unknownfunctionsand theirderivatives.^[1]In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines includingengineering,physics,economics,andbiology.

The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

Often when aclosed-form expressionfor the solutions is not available, solutions may be approximated numerically using computers. Thetheory of dynamical systemsputs emphasis onqualitativeanalysis of systems described by differential equations, while manynumerical methodshave been developed to determine solutions with a given degree of accuracy.

History

Differential equations came into existence with theinvention of calculusbyIsaac NewtonandGottfried Leibniz.In Chapter 2 of his 1671 workMethodus fluxionum et Serierum Infinitarum,^[2]Newton listed three kinds of differential equations:

{\begin{aligned}{\frac {dy}{dx}}&=f(x)\\[4pt]{\frac {dy}{dx}}&=f(x,y)\\[4pt]x_{1}{\frac {\partial y}{\partial x_{1}}}&+x_{2}{\frac {\partial y}{\partial x_{2}}}=y\end{aligned}}

In all these cases, $y$ is an unknown function of $x$ (or of $x 1$ and $x 2$ ), and $f$ is a given function.

He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

Jacob Bernoulliproposed theBernoulli differential equationin 1695.^[3]This is anordinary differential equationof the form

y'+P(x)y=Q(x)y^{n}\,

for which the following year Leibniz obtained solutions by simplifying it.^[4]

Historically, the problem of a vibrating string such as that of amusical instrumentwas studied byJean le Rond d'Alembert,Leonhard Euler,Daniel Bernoulli,andJoseph-Louis Lagrange.^[5]^[6]^[7]^[8]In 1746, d’Alembert discovered the one-dimensionalwave equation,and within ten years Euler discovered the three-dimensional wave equation.^[9]

TheEuler–Lagrange equationwas developed in the 1750s by Euler and Lagrange in connection with their studies of thetautochroneproblem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it tomechanics,which led to the formulation ofLagrangian mechanics.

In 1822,Fourierpublished his work onheat flowinThéorie analytique de la chaleur(The Analytic Theory of Heat),^[10]in which he based his reasoning onNewton's law of cooling,namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of hisheat equationfor conductive diffusion of heat. This partial differential equation is now a common part of mathematical physics curriculum.

Example

Inclassical mechanics,the motion of a body is described by its position and velocity as the time value varies.Newton's lawsallow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

In some cases, this differential equation (called anequation of motion) may be solved explicitly.

An example of modeling 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 a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

Types

Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

Ordinary differential equations

Anordinary differential equation(ODE) is an equation containing an unknownfunction of one real or complex variable $x$ ,its derivatives, and some given functions of $x$ .The unknown function is generally represented by avariable(often denoted $y$ ), which, therefore,dependson $x$ .Thus $x$ is often called theindependent variableof the equation. The term "ordinary"is used in contrast with the termpartial differential equation,which may be with respect tomore thanone independent variable.

Linear differential equationsare the differential equations that arelinearin the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms ofintegrals.

Most ODEs that are encountered inphysicsare linear. Therefore, mostspecial functionsmay be defined as solutions of linear differential equations (seeHolonomic function).

As, in general, the solutions of a differential equation cannot be expressed by aclosed-form expression,numerical methodsare commonly used for solving differential equations on a computer.

Partial differential equations

Apartial differential equation(PDE) is a differential equation that contains unknownmultivariable functionsand theirpartial derivatives.(This is in contrast toordinary differential equations,which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevantcomputer model.

PDEs can be used to describe a wide variety of phenomena in nature such assound,heat,electrostatics,electrodynamics,fluid flow,elasticity,orquantum mechanics.These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensionaldynamical systems,partial differential equations often modelmultidimensional systems.Stochastic partial differential equationsgeneralize partial differential equations for modelingrandomness.

Non-linear differential equations

Anon-linear differential equationis a differential equation that is not alinear equationin the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particularsymmetries.Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic ofchaos.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). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.^[11]

Linear differential equations frequently appear asapproximationsto nonlinear equations. These approximations are only valid under restricted conditions. For example, theharmonic oscillatorequation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.

Equation order and degree

Theorder of the differential equationis the highestorder of derivativeof the unknown function that appears in the differential equation. For example, an equation containing onlyfirst-order derivativesis afirst-order differential equation,an equation containing thesecond-order derivativeis asecond-order differential equation,and so on.^[12]^[13]

When it is written as apolynomial equationin the unknown function and its derivatives, itsdegree of the differential equationis, depending on the context, thepolynomial degreein the highest derivative of the unknown function,^[14]or itstotal degreein the unknown function and its derivatives. In particular, alinear differential equationhas degree one for both meanings, but the non-linear differential equation $y'+y^{2}=0$ is of degree one for the first meaning but not for the second one.

Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as thethin-film equation,which is a fourth order partial differential equation.

Examples

In the first group of examplesuis an unknown function ofx,andcandωare constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing betweenlinearandnonlineardifferential equations, and betweenhomogeneousdifferential equationsandheterogeneousones.

Heterogeneous first-order linear constant coefficient ordinary differential equation:
${\frac {du}{dx}}=cu+x^{2}.$
Homogeneous second-order linear ordinary differential equation:
${\frac {d^{2}u}{dx^{2}}}-x{\frac {du}{dx}}+u=0.$
Homogeneous second-order linear constant coefficient ordinary differential equation describing theharmonic oscillator:
${\frac {d^{2}u}{dx^{2}}}+\omega ^{2}u=0.$
Heterogeneous first-order nonlinear ordinary differential equation:
${\frac {du}{dx}}=u^{2}+4.$
Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of apendulumof lengthL:
$L{\frac {d^{2}u}{dx^{2}}}+g\sin u=0.$

In the next group of examples, the unknown functionudepends on two variablesxandtorxandy.

Homogeneous first-order linear partial differential equation:
${\frac {\partial u}{\partial t}}+t{\frac {\partial u}{\partial x}}=0.$
Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, theLaplace equation:
${\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.$
Homogeneous third-order non-linear partial differential equation, theKdV equation:
${\frac {\partial u}{\partial t}}=6u{\frac {\partial u}{\partial x}}-{\frac {\partial ^{3}u}{\partial x^{3}}}.$

Existence of solutions

Solvingdifferential equations is not like solvingalgebraic equations.Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

For first order initial value problems, thePeano existence theoremgives one set of circumstances in which a solution exists. Given any point $(a,b)$ in the xy-plane, define some rectangular region $Z$ ,such that $Z=[l,m]\times [n,p]$ and $(a,b)$ is in the interior of $Z$ .If we are given a differential equation ${\textstyle {\frac {dy}{dx}}=g(x,y)}$ and the condition that $y=b$ when $x=a$ ,then there is locally a solution to this problem if $g(x,y)$ and ${\textstyle {\frac {\partial g}{\partial x}}}$ are both continuous on $Z$ .This solution exists on some interval with its center at $a$ .The solution may not be unique. (SeeOrdinary differential equationfor other results.)

However, this only helps us with first orderinitial value problems.Suppose we had a linear initial value problem of the nth order:

f_{n}(x){\frac {d^{n}y}{dx^{n}}}+\cdots +f_{1}(x){\frac {dy}{dx}}+f_{0}(x)y=g(x)

such that

{\begin{aligned}y(x_{0})&=y_{0},&y'(x_{0})&=y'_{0},&y''(x_{0})&=y''_{0},&\ldots \end{aligned}}

For any nonzero $f_{n}(x)$ ,if $\{f_{0},f_{1},\ldots \}$ and $g$ are continuous on some interval containing $x_{0}$ , $y$ exists and is unique.^[15]

Related concepts

Adelay 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.
Integral equationsmay be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation containsintegrals.
Anintegro-differential equation(IDE) is an equation that combines aspects of a differential equation and an integral equation.
Astochastic differential equation(SDE) is an equation in which the unknown quantity is astochastic processand the equation involves some known stochastic processes, for example, theWiener processin the case of diffusion equations.
Astochastic partial differential equation(SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications inquantum field theoryandstatistical mechanics.
An ultrametricpseudo-differential equationis an equation which containsp-adic numbersin anultrametric space.Mathematical models that involve ultrametric pseudo-differential equations usepseudo-differential operatorsinstead ofdifferential operators.
Adifferential algebraic equation(DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

Connection to difference equations

The theory of differential equations is closely related to the theory ofdifference 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 the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

Applications

The study of differential equations is a wide field inpureandapplied mathematics,physics,andengineering.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 modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not haveclosed formsolutions. Instead, solutions can be approximated usingnumerical methods.

Many fundamental laws ofphysicsandchemistrycan be formulated as differential equations. Inbiologyandeconomics,differential equations are used tomodelthe 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 the 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,thewave 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 byJoseph Fourier,is governed by another second-order partial differential equation, theheat equation.It turns out that manydiffusionprocesses, while seemingly different, are described by the same equation; theBlack–Scholesequation in finance is, for instance, related to the heat equation.

The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. SeeList of named differential equations.

Software

SomeCASsoftware can solve differential equations. These are the commands used in the leading programs:

Maple:^[16]dsolve
Mathematica:^[17]DSolve[]
Maxima:^[18]ode2(equation, y, x)
SageMath:^[19]desolve()
SymPy:^[20]sympy.solvers.ode.dsolve(equation)
Xcas:^[21]desolve(y'=k*y,y)

References

^Dennis G. Zill (15 March 2012).A First Course in Differential Equations with Modeling Applications.Cengage Learning.ISBN 978-1-285-40110-2.
^Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].
^Bernoulli, Jacob(1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis",Acta Eruditorum
^Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993),Solving ordinary differential equations I: Nonstiff problems,Berlin, New York:Springer-Verlag,ISBN 978-3-540-56670-0
^Frasier, Craig (July 1983)."Review ofThe evolution of dynamics, vibration theory from 1687 to 1742,by John T. Cannon and Sigalia Dostrovsky "(PDF).Bulletin of the American Mathematical Society.New Series.9(1).
^Wheeler, Gerard F.; Crummett, William P. (1987). "The Vibrating String Controversy".Am. J. Phys.55(1): 33–37.Bibcode:1987AmJPh..55...33W.doi:10.1119/1.15311.
^For a special collection of the 9 groundbreaking papers by the three authors, seeFirst Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings Archived2020-02-09 at theWayback Machine(retrieved 13 Nov 2012). Herman HJ Lynge and Son.
^For de Lagrange's contributions to the acoustic wave equation, can consultAcoustics: An Introduction to Its Physical Principles and ApplicationsAllan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)
^Speiser, David.Discovering the Principles of Mechanics 1600-1800,p. 191 (Basel: Birkhäuser, 2008).
^Fourier, Joseph (1822).Théorie analytique de la chaleur(in French). Paris: Firmin Didot Père et Fils.OCLC 2688081.
^Boyce, William E.; DiPrima, Richard C. (1967).Elementary Differential Equations and Boundary Value Problems(4th ed.). John Wiley & Sons. p. 3.
^Weisstein, Eric W."Ordinary Differential Equation Order." FromMathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html
^Order and degree of a differential equation Archived2016-04-01 at theWayback Machine,accessed Dec 2015.
^Elias Loomis (1887).Elements of the Differential and Integral Calculus(revised ed.). Harper & Bros. p. 247.Extract of page 247
^Zill, Dennis G. (2001).A First Course in Differential Equations(5th ed.). Brooks/Cole.ISBN 0-534-37388-7.
^"dsolve - Maple Programming Help".www.maplesoft.com.Retrieved2020-05-09.
^"DSolve - Wolfram Language Documentation".www.wolfram.com.Retrieved2020-06-28.
^Schelter, William F.Gaertner, Boris (ed.)."Differential Equations - Symbolic Solutions".The Computer Algebra Program Maxima - a Tutorial (in Maxima documentation onSourceForge).Archivedfrom the original on 2022-10-04.
^"Basic Algebra and Calculus — Sage Tutorial v9.0".doc.sagemath.org.Retrieved2020-05-09.
^"ODE".SymPy 1.11 documentation.2022-08-22.Archivedfrom the original on 2022-09-26.
^"Symbolic algebra and Mathematics with Xcas"(PDF).

External links

Media related toDifferential equationsat Wikimedia Commons
Lectures on Differential Equations MITOpen CourseWare Videos
Online Notes / Differential EquationsPaul Dawkins,Lamar University
Differential Equations,S.O.S. Mathematics
Introduction to modeling via differential equationsIntroduction to modeling by means of differential equations, with critical remarks.
Mathematical Assistant on WebSymbolic ODE tool, usingMaxima
Exact Solutions of Ordinary Differential Equations
Collection of ODE and DAE models of physical systems Archived2008-12-19 at theWayback MachineMATLAB models
Notes on Diffy Qs: Differential Equations for EngineersAn introductory textbook on differential equations by Jiri Lebl ofUIUC
Khan Academy Video playlist on differential equationsTopics covered in a first year course in differential equations.
MathDiscuss Video playlist on differential equations

[Zill2012-1] Dennis G. Zill (15 March 2012).A First Course in Differential Equations with Modeling Applications.Cengage Learning.ISBN 978-1-285-40110-2.

[2] Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].

[3] Bernoulli, Jacob(1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis",Acta Eruditorum

[4] Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993),Solving ordinary differential equations I: Nonstiff problems,Berlin, New York:Springer-Verlag,ISBN 978-3-540-56670-0

[5] Frasier, Craig (July 1983)."Review ofThe evolution of dynamics, vibration theory from 1687 to 1742,by John T. Cannon and Sigalia Dostrovsky "(PDF).Bulletin of the American Mathematical Society.New Series.9(1).

[6] Wheeler, Gerard F.; Crummett, William P. (1987). "The Vibrating String Controversy".Am. J. Phys.55(1): 33–37.Bibcode:1987AmJPh..55...33W.doi:10.1119/1.15311.

[7] For a special collection of the 9 groundbreaking papers by the three authors, seeFirst Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings Archived2020-02-09 at theWayback Machine(retrieved 13 Nov 2012). Herman HJ Lynge and Son.

[8] For de Lagrange's contributions to the acoustic wave equation, can consultAcoustics: An Introduction to Its Physical Principles and ApplicationsAllan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)

[Speiser-9] Speiser, David.Discovering the Principles of Mechanics 1600-1800,p. 191 (Basel: Birkhäuser, 2008).

[10] Fourier, Joseph (1822).Théorie analytique de la chaleur(in French). Paris: Firmin Didot Père et Fils.OCLC 2688081.

[11] Boyce, William E.; DiPrima, Richard C. (1967).Elementary Differential Equations and Boundary Value Problems(4th ed.). John Wiley & Sons. p. 3.

[12] Weisstein, Eric W."Ordinary Differential Equation Order." FromMathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html

[13] Order and degree of a differential equation Archived2016-04-01 at theWayback Machine,accessed Dec 2015.

[14] Elias Loomis (1887).Elements of the Differential and Integral Calculus(revised ed.). Harper & Bros. p. 247.Extract of page 247

[15] Zill, Dennis G. (2001).A First Course in Differential Equations(5th ed.). Brooks/Cole.ISBN 0-534-37388-7.

[16] "dsolve - Maple Programming Help".www.maplesoft.com.Retrieved2020-05-09.

[17] "DSolve - Wolfram Language Documentation".www.wolfram.com.Retrieved2020-06-28.

[18] Schelter, William F.Gaertner, Boris (ed.)."Differential Equations - Symbolic Solutions".The Computer Algebra Program Maxima - a Tutorial (in Maxima documentation onSourceForge).Archivedfrom the original on 2022-10-04.

[19] "Basic Algebra and Calculus — Sage Tutorial v9.0".doc.sagemath.org.Retrieved2020-05-09.

[20] "ODE".SymPy 1.11 documentation.2022-08-22.Archivedfrom the original on 2022-09-26.

[21] "Symbolic algebra and Mathematics with Xcas"(PDF).

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