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Differential equation

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   In mathematics, a differential equation is an equation in which the
   derivatives of a function appear as variables. Many of the fundamental
   laws of physics, chemistry, biology and economics can be formulated as
   differential equations. They express the relationship involving the
   rates of change of continuously changing quantities modeled by
   functions and are used whenever a rate of change (the derivative) is
   known but the process originating is not. The solution of a
   differential equation is usually a function whose derivatives satisfy
   the equation. The question then becomes how to find the solutions of
   those equations.

   The mathematical theory of differential equations has developed
   together with the sciences where the equations originate and where the
   results find application. Diverse scientific fields often give rise to
   identical problems in differential equations. In such cases, the
   mathematical theory can unify otherwise quite distinct scientific
   fields. A celebrated example is Fourier's theory of the conduction of
   heat in terms of sums of trigonometric functions, Fourier series, which
   finds application in the propagation of sound and electromagnetic
   fields, optics, elasticity, spectral analysis of radiation, and other
   scientific fields.

   The order of a differential equation is that of the highest derivative
   that it contains. For instance, a first-order differential equation
   contains only first derivatives.

Types of differential equations

     * An ordinary differential equation (ODE) only contains functions of
       one independent variable, and derivatives in that variable.
     * A partial differential equation (PDE) contains functions of
       multiple independent variables and their partial derivatives.
     * A delay differential equation (DDE) contains functions of one
       dependent variable, derivatives in that variable, and depends on
       previous states of the dependent variables.
     * A stochastic differential equation (SDE) is a differential equation
       in which one or more of the terms is a stochastic process, thus
       resulting in a solution which is itself a stochastic process.
     * A differential algebraic equation (DAE) is a differential equation
       comprising differential and algebraic terms, given in implicit
       form.

   Each of those categories is divided into linear and nonlinear
   subcategories. A differential equation is linear if it involves the
   unknown function and its derivatives only to the first power; otherwise
   the differential equation is nonlinear. Thus if u' denotes the first
   derivative of u, then the equation

          u' = u

   is linear, while the equation

          u' = u^2

   is nonlinear. Solutions of a linear equation in which the unknown
   function or its derivative or derivatives appear in each term (linear
   homogeneous equations) may be added together or multiplied by an
   arbitrary constant in order to obtain additional solutions of that
   equation, but there is no general way to obtain families of solutions
   of nonlinear equations, except when they exhibit symmetries; see
   symmetries and invariants. Linear equations frequently appear as
   approximations to nonlinear equations, and these approximations are
   only valid under restricted conditions.

   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.

   The study of differential equations is a wide field in both pure and
   applied mathematics. Pure mathematicians study the types and properties
   of differential equations, such as whether or not solutions exist, and
   should they exist, whether they are unique. Applied mathematicians
   emphasize differential equations from applications, and in addition to
   existence/uniqueness questions, are also concerned with rigorously
   justifying methods for approximating solutions. Physicists and
   engineers are usually more interested in computing approximate
   solutions to differential equations, and are typically less interested
   in justifications for whether these approximations really are close to
   the actual solutions. These solutions are then used to simulate
   celestial motions, simulate neurons, design bridges, automobiles,
   aircraft, sewers, etc. Often, these equations do not have closed form
   solutions and are solved 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 shocks in hyperbolic (or wave) equations.

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

Famous differential equations

     * Newton's Second Law in dynamics (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 Schrödinger equation in quantum mechanics
     * The geodesic equation
     * The Navier-Stokes equations in fluid dynamics
     * The Lotka-Volterra equation in population dynamics
     * The Black-Scholes equation in finance
     * The Cauchy-Riemann equations in complex analysis

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