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Differential equation

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Differential equations 

Navier–Stokes differential equations used to simulate airflow around an obstruction.

Classification 
Solution 
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 manynumerical methods have been developed to determine solutions with a given degree of accuracy.
Contents
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History[edit]
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: those involving two derivatives (or fluxions) and only one undifferentiated quantity ; those involving and ; and those involving more than two derivatives. As examples of the three cases, he solves the equations:

 ,

 , and

 , respectively.
He solves these examples and others using infinite series and discusses the nonuniqueness of solutions.
Jacob Bernoulli solved the Bernoulli differential equation in 1695.^{[2]} This is an ordinary differential equation of the form
for which he obtained exact solutions.^{[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 ofLagrangian mechanics.
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 hisheat equation for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.
Example[edit]
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 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 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.
Main topics[edit]
Ordinary differential equations[edit]
An ordinary differential equation or ODE is an equation containing a function of one independent variable and its derivatives. The term "ordinary" is used in contrast with the termpartial differential equation which may be with respect to more than one independent variable.
Linear differential equations, which have solutions that can be added and multiplied by coefficients, are welldefined and understood, and exact closedform solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.
Partial differential equations[edit]
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 by hand, or used to create a relevant computer model.
PDEs can be used to describe a wide variety of phenomena 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.
Linear and nonlinear[edit]
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 of the unknown function and its derivatives are not allowed) andnonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an 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 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).
Examples[edit]
In the first group of examples, let u be an unknown function of x, and c and ω are known constants.
 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:
 Thirdorder nonlinear partial differential equation, the Korteweg–de Vries equation:
Existence of Solutions[edit]
Solving differential equations is not like solving algebraic equations. Not only are their solutions often times unclear, but whether solutions are unique or exist at all are also notable subjects of interest.
For first order initial value problems, it is easy to tell whether a unique solution exists. Given any point in the xyplane, define some rectangular region , such that and is in . If we are given a differential equation and an initial condition , then there is a unique solution to this initial value problem if and are both continuous on . This unique solution exists on some interval with its center at .
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.
^{[11]}
Related concepts[edit]
 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[edit]
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.
Applications and connections to other areas[edit]
In general[edit]
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 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.
In physics[edit]
 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, andelectric 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.^{[12]} First published by Einstein in 1915^{[13]} 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).^{[14]}
 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'swave function (also called a "state function").^{[15]}^{:1–2}
 Other important equations
 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.
In biology[edit]
 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 dynamicsof biological systems in which two species interact, one as a predator and the other as prey.
 Other important equations
 Verhulst equation – biological population growth
 von Bertalanffy model – biological individual growth
 Replicator dynamics – found in theoretical biology
 Hodgkin–Huxley model – neural action potentials
In chemistry[edit]
 Rate equation
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).^{[16]} To determine the rate equation for a particular system one combines the reaction rate with a mass balance for the system.^{[17]}
In economics[edit]
 Important equations
 The key equation of the Solow–Swan model is:
 The Black–Scholes PDE
 Malthusian growth model
 The Vidale–Wolfe advertising model
See also[edit]
 Complex differential equation
 Exact differential equation
 Initial condition
 Integral equations
 Numerical methods
 Picard–Lindelöf theorem on existence and uniqueness of solutions
 Recurrence relation, also known as 'Difference Equation'