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Differential equation

From Wikipedia, the free encyclopedia

Type of functional equation (mathematics)

Not to be confused withDifference equation.

Differential equations

Scope

Fields

Natural sciences Engineering
Astronomy Physics Chemistry Biology Geology
Applied mathematics
Continuum mechanics Chaos theory Dynamical systems
Social sciences
Economics Population dynamics

List of named differential equations

Classification

Types

By variable type
Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear
Dependent and independent variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous
Features
Order Operator Notation

Relation to processes

Difference(discrete analogue)

Solution

Existence and uniqueness

General topics

Initial conditions
Boundary values
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function

Solution methods

People

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Inmathematics, adifferential equation is anequation that relates one or more unknownfunctions and 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 inmathematical models andscientific laws; 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 expression for the solutions is not available, solutions may be approximated numerically using computers, and manynumerical methods have been developed to determine solutions with a given degree of accuracy. Thetheory of dynamical systems analyzes thequalitative aspects of solutions, such as theiraverage behavior over a long time interval.

History

Differential equations came into existence with theinvention of calculus byIsaac Newton andGottfried 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 ofx (or ofx₁ andx₂), andf is a given function.

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

Jacob Bernoulli proposed theBernoulli differential equation in 1695.^[3] This is anordinary differential equation of 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 instrument was 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 equation was developed in the 1750s by Euler and Lagrange in connection with their studies of thetautochrone problem. 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,Fourier published his work onheat flow inThéorie 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 equation for 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 laws allow 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 classified several different ways. Besides 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

Main articles:Ordinary differential equation andLinear differential equation

Anordinary differential equation (ODE) is an equation containing an unknownfunction of one real or complex variablex, its derivatives, and some given functions ofx. The unknown function is generally represented by avariable (often denotedy), which, therefore,depends onx. Thusx is often called theindependent variable of the equation. The term "ordinary" is used in contrast with the termpartial differential equation, which may be with respect tomore than one independent variable.

Linear differential equations are the differential equations that arelinear in 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 inphysics are linear. Therefore, mostspecial functions may 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 methods are commonly used for solving differential equations on a computer.

Partial differential equations

Main article:Partial differential equation

Apartial differential equation (PDE) is a differential equation that contains unknownmultivariable functions and 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 equations generalize partial differential equations for modelingrandomness.

Non-linear differential equations

Main article:Non-linear differential equations

Anon-linear differential equation is a differential equation that is not alinear equation in 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 asapproximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, theharmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.

Equation order and degree

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

When it is written as apolynomial equation in the unknown function and its derivatives, itsdegree of the differential equation is, depending on the context, thepolynomial degree in the highest derivative of the unknown function,^[14] or itstotal degree in the unknown function and its derivatives. In particular, alinear differential equation has 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 examplesu is an unknown function ofx, andc andω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing betweenlinear andnonlinear differential equations, and betweenhomogeneous differential equations andheterogeneous ones.

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 apendulum of lengthL:
$L{\frac {d^{2}u}{dx^{2}}}+g\sin u=0.$

In the next group of examples, the unknown functionu depends on two variablesx andt orx andy.

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

Solving differential 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 theorem gives one set of circumstances in which a solution exists. Given any point $(a,b)$ in the xy-plane, define some rectangular region $Z {\displaystyle Z}$ , such that $Z=[l,m]\times [n,p]$ and $(a,b)$ is in the interior of $Z {\displaystyle Z}$ . 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 $Z {\displaystyle Z}$ . This solution exists on some interval with its center at $a {\displaystyle a}$ . The solution may not be unique. (SeeOrdinary differential equation for 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 $g {\displaystyle g}$ are continuous on some interval containing $x_{0}$ , $y {\displaystyle y}$ 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 equations may 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 process and the equation involves some known stochastic processes, for example, theWiener process in 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 theory andstatistical mechanics.
An ultrametricpseudo-differential equation is an equation which containsp-adic numbers in anultrametric space. Mathematical models that involve ultrametric pseudo-differential equations usepseudo-differential operators instead ofdifferential operators.
Adifferential 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 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 inpure andapplied 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 form solutions. Instead, solutions can be approximated usingnumerical methods.

Many fundamental laws ofphysics andchemistry can be formulated as differential equations. Inbiology andeconomics, differential equations are used tomodel the behavior ofcomplex 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 manydiffusion processes, while seemingly different, are described by the same equation; theBlack–Scholes equation 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

SomeCAS software 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)

See also

Exact differential equation
Functional differential equation
Initial condition
Integral equations
Numerical methods for ordinary differential equations
Numerical methods for partial differential equations
Picard–Lindelöf theorem on existence and uniqueness of solutions
Recurrence relation, also known as 'difference equation'
Abstract differential equation
System of differential equations

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 Archived 2020-02-09 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 Applications Allan 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, 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).Archived from 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.Archived from the original on 2022-09-26.
^"Symbolic algebra and Mathematics with Xcas"(PDF).

Further reading

Abbott, P.; Neill, Hugh (2003).Calculus. London ; Chicago: Hodder & Stoughton Educational ;Contemporary Books. pp. 266–277.ISBN 978-0-07-142128-7.
Blanchard, Paul (2006).Student Solutions Manual for Blanchard, Devaney, and Hall's Differential Equations (3rd ed.). Belmont, CA: Brooks/Cole Thomson Learning.ISBN 978-0-495-01461-4.
Boyce, William E.;DiPrima, Richard C.; Meade, Douglas B. (2017).Elementary Differential Equations and Boundary Value Problems (11th ed.). Hoboken, NJ: John Wiley & Sons.ISBN 978-1-119-44376-6.
Coddington, E. A.;Levinson, N. (1955).Theory of Ordinary Differential Equations. New York: McGraw-Hill.
Ince, E. L. (1956).Ordinary Differential Equations. New York: Dover.
Johnson, W. (1913).A Treatise on Ordinary and Partial Differential Equations. John Wiley and Sons. InUniversity of Michigan Historical Math Collection
Polyanin, A. D.; Zaitsev, V. F. (2003).Handbook of Exact Solutions for Ordinary Differential Equations (2nd ed.). Boca Raton: Chapman & Hall/CRC Press.ISBN 1-58488-297-2.
Porter, Ronald I. (1978). "XIX Differential Equations".Further elementary analysis (4th ed.). London:Bell & Hyman.ISBN 978-0-7135-1594-7.
Teschl, Gerald (2012).Ordinary Differential Equations and Dynamical Systems. Providence, RI:American Mathematical Society.ISBN 978-0-8218-8328-0.
Zwillinger, Daniel (2014).Handbook of Differential Equations (2nd ed.). Burlington:Elsevier Science.ISBN 978-1-4832-6396-0.

External links

Wikiquote has quotations related toDifferential equation.

Wikibooks has a book on the topic of:Ordinary Differential Equations

Wikiversity has learning resources aboutDifferential equations

Wikisource has the text of the1911Encyclopædia Britannica article "Differential Equation".

Media related toDifferential equations at Wikimedia Commons
Lectures on Differential Equations MIT Open CourseWare Videos
Online Notes / Differential Equations Paul Dawkins,Lamar University
Differential Equations, S.O.S. Mathematics
Introduction to modeling via differential equations Introduction to modeling by means of differential equations, with critical remarks.
Mathematical Assistant on Web Symbolic ODE tool, usingMaxima
Exact Solutions of Ordinary Differential Equations
Collection of ODE and DAE models of physical systems Archived 2008-12-19 at theWayback Machine MATLAB models
Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl ofUIUC
Khan Academy Video playlist on differential equations Topics covered in a first year course in differential equations.
MathDiscuss Video playlist on differential equations

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Differential equations

Classification

Operations	Differential operator Notation for differentiation Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear Holonomic
Attributes of variables	Dependent and independent variables Homogeneous Nonhomogeneous Coupled Decoupled Order Degree Autonomous Exact differential equation On jet bundles
Relation to processes	Difference (discrete analogue) Stochastic Stochastic partial Delay

Solutions

Existence/uniqueness	Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem
Solution topics	Wronskian Phase portrait Phase space Lyapunov stability Asymptotic stability Exponential stability Rate of convergence Series solutions Integral solutions Numerical integration Dirac delta function
Solution methods	Inspection Substitution Separation of variables Method of undetermined coefficients Variation of parameters Integrating factor Integral transforms Euler method Finite difference method Crank–Nicolson method Runge–Kutta methods Finite element method Finite volume method Galerkin method Perturbation theory

Examples

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Differential equations

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