Differential equation
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.
If a selfcontained 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.
Contents
History
Differential equations first came into existence with the invention of calculus by Newton and Leibniz. In Chapter 2 of his 1671 work "Methodus fluxionum et Serierum Infinitarum",^{[1]} Isaac Newton listed three kinds of differential equations:
He solves these examples and others using infinite series and discusses the nonuniqueness of solutions.
Jacob Bernoulli proposed the Bernoulli differential equation in 1695.^{[2]} This is an ordinary differential equation of the form
for which the following year Leibniz obtained solutions by simplifying it.^{[3]}
Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and JosephLouis Lagrange.^{[4]}^{[5]}^{[6]}^{[7]} In 1746, d’Alembert discovered the onedimensional wave equation, and within ten years Euler discovered the threedimensional wave equation.^{[8]}
The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone 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 to mechanics, which led to the formulation of Lagrangian mechanics.
In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat),^{[9]} in which he based his reasoning on Newton'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 his heat equation for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.
Example
For example, in classical 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 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 acceleration 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/Partial, Linear/Nonlinear, and Homogeneous/Inhomogeneous. 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
An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.
Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and, in many cases, one may express their solutions in terms of integrals.
Most ODEs that are encountered in physics are linear, and, therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function).
As, in general, the solutions of a differential equation cannot be expressed by a closedform expression, numerical methods are commonly used for solving differential equations on a computer.
Partial differential equations
A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary 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 relevant computer model.
PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model onedimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
Nonlinear differential equations
Nonlinear differential equations are formed by the products of the unknown function and its derivatives are allowed and its degree is > 1. There are very few methods of solving nonlinear differential equations exactly; 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 wellposedness 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.^{[10]}
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).
Equation order
Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a firstorder differential equation, an equation containing the second derivative is a secondorder differential equation, and so on.^{[11]}^{[12]} Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation.
Examples
In the first group of examples, u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consists of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and inhomogeneous ones.
 Inhomogeneous firstorder linear constant coefficient ordinary differential equation:
 Homogeneous secondorder linear ordinary differential equation:
 Homogeneous secondorder linear constant coefficient ordinary differential equation describing the harmonic oscillator:
 Inhomogeneous firstorder nonlinear ordinary differential equation:
 Secondorder nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:
In the next group of examples, the unknown function u depends on two variables x and t or x and y.
 Homogeneous firstorder linear partial differential equation:
 Homogeneous secondorder linear constant coefficient partial differential equation of elliptic type, the Laplace equation:
 Homogeneous thirdorder nonlinear partial differential equation :
Existence of solutions
Solving differential equations is not like solving algebraic 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, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point in the xyplane, define some rectangular region , such that and is in the interior of . If we are given a differential equation and the condition that when , then there is locally a solution to this problem if and are both continuous on . This solution exists on some interval with its center at . The solution may not be unique. (See Ordinary differential equation for other results.)
However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:
such that
For any nonzero , if and are continuous on some interval containing , is unique and exists.^{[13]}
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 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
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 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 in pure and applied mathematics, physics, 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 interactions between neurons. Differential equations such as those used to solve reallife problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.
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 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 secondorder 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 secondorder partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.
Physics
 Euler–Lagrange equation in classical mechanics
 Hamilton's equations in classical mechanics
 Radioactive decay in nuclear physics
 Newton's law of cooling in thermodynamics
 The wave equation
 The heat equation in thermodynamics
 Laplace's equation, which defines harmonic functions
 Poisson's equation
 The geodesic equation
 The Navier–Stokes equations in fluid dynamics
 The Diffusion equation in stochastic processes
 The Convection–diffusion equation in fluid dynamics
 The Cauchy–Riemann equations in complex analysis
 The Poisson–Boltzmann equation in molecular dynamics
 The shallow water equations
 Universal differential equation
 The Lorenz equations whose solutions exhibit chaotic flow.
 The Sagnik's equation in mechanics
Classical mechanics
So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion.
Electrodynamics
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the Scottish physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862.
General relativity
The Einstein field equations (EFE; also known as "Einstein's equations") are a set of ten partial differential equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.^{[14]} First published by Einstein in 1915^{[15]} as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).^{[16]}
Quantum mechanics
In quantum mechanics, the analogue of Newton's law is Schrödinger's equation (a partial differential equation) for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the timeevolution of the system's wave function (also called a "state function").^{[17]}
Biology
 Verhulst equation – biological population growth
 von Bertalanffy model – biological individual growth
 Replicator dynamics – found in theoretical biology
 Hodgkin–Huxley model – neural action potentials
Predatorprey equations
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of firstorder, nonlinear, differential equations frequently used to describe the population dynamics of two species that interact, one as a predator and the other as prey.
Chemistry
The rate law or rate equation for a chemical reaction is a differential equation that links the reaction rate with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders).^{[18]} To determine the rate equation for a particular system one combines the reaction rate with a mass balance for the system.^{[19]} In addition, a range of differential equations are present in the study of thermodynamics and quantum mechanics.
Economics
 The key equation of the Solow–Swan model is
 The Black–Scholes PDE
 Malthusian growth model
 The Vidale–Wolfe advertising model
See also
 Complex differential equation
 Exact differential equation
 Functional differential equation
 Initial condition
 Integral equations
 Numerical methods for solving differential equations
 Picard–Lindelöf theorem on existence and uniqueness of solutions
 Recurrence relation, also known as 'difference equation'
 Abstract differential equation
References
 ^ 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: SpringerVerlag, ISBN 9783540566700
 ^ Cannon, John T.; Dostrovsky, Sigalia (1981). "The evolution of dynamics, vibration theory from 1687 to 1742". Studies in the History of Mathematics and Physical Sciences. 6. New York: SpringerVerlag: ix + 184 pp. ISBN 0387906266. GRAY, JW (July 1983). "BOOK REVIEWS". Bulletin (New Series) of the American Mathematical Society. 9 (1). (retrieved 13 Nov 2012).
 ^ 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, see First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli.  the controversy about vibrating strings (retrieved 13 Nov 2012). Herman HJ Lynge and Son.
 ^ For de Lagrange's contributions to the acoustic wave equation, can consult Acoustics: 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 16001800, 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." From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html
 ^ Order and degree of a differential equation, accessed Dec 2015.
 ^ Zill, Dennis G. A First Course in Differential Equations (5th ed.). Brooks/Cole. ISBN 0534373887.
 ^ Einstein, Albert (1916). "The Foundation of the General Theory of Relativity". Annalen der Physik. 354 (7): 769. Bibcode:1916AnP...354..769E. doi:10.1002/andp.19163540702. Archived from the original (PDF) on 20060829.
 ^ Einstein, Albert (November 25, 1915). "Die Feldgleichungen der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847. Retrieved 20060912.
 ^ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 9780716703440 Chapter 34, p. 916.
 ^ Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, pp. 1–2, ISBN 0131118927
 ^ IUPAC Gold Book definition of rate law. See also: According to IUPAC Compendium of Chemical Terminology.
 ^ Kenneth A. Connors Chemical Kinetics, the study of reaction rates in solution, 1991, VCH Publishers.
Further reading
 Abbott, P.; Neill, H. (2003). Teach Yourself Calculus. pp. 266–277.
 Blanchard, P.; Devaney, R. L.; Hall, G. R. (2006). Differential Equations. Thompson.
 Coddington, E. A.; Levinson, N. (1955). Theory of Ordinary Differential Equations. McGrawHill.
 Ince, E. L. (1956). Ordinary Differential Equations. Dover.
 Johnson,, W. (1913). A Treatise on Ordinary and Partial Differential Equations. John Wiley and Sons. In University 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 1584882972.
 Porter, R. I. (1978). "XIX Differential Equations". Further Elementary Analysis.
 Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 9780821883280.
 Zwillinger, D. (1997). Handbook of Differential Equations (3rd ed.). Boston: Academic Press.
External links
Wikiquote has quotations related to: Differential equation 
Wikibooks has a book on the topic of: Ordinary Differential Equations 
Wikiversity has learning resources about Differential equations 
Wikisource has the text of the 1911 Encyclopædia Britannica article Differential Equation. 
 Media related to Differential 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, using Maxima
 Exact Solutions of Ordinary Differential Equations
 Collection of ODE and DAE models of physical systems MATLAB models
 Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl of UIUC
 Khan Academy Video playlist on differential equations Topics covered in a first year course in differential equations.
 MathDiscuss Video playlist on differential equations