A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics and other disciplines.
Differential equations arise in many areas of science and technology; whenever a deterministic relationship involving some continuously changing quantities (modeled by functions) and their rates of change (expressed as derivatives) is known or postulated. This is well illustrated by classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's Laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In many cases, this differential equation may be solved explicitly, yielding the law of motion.
Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions, functions that make the equation hold true. Only the simplest differential equations admit solutions given by explicit formulas. Many properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained 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.
Directions of study
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 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 have closed form solutions. Instead, solutions can be approximated 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 possible presence of shocks for equations of hyperbolic type.
The study of the stability of solutions of differential equations is known as stability theory.
Types of differential equations
- An ordinary differential equation (ODE) is a differential equation in which the unknown function is a function of a single independent variable.
- A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and their partial derivatives.
- A delay differential equation (DDE) is a differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
- 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 the dependent variable and all its derivatives appear to the power 1 and there are no products or functions of the dependent variable. Otherwise the differential equation is nonlinear. Thus if u′ denotes the first derivative of the function u, then the equation
is linear, while the equation
is nonlinear. Solutions of a linear equation in which the unknown function or its derivative(s) 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.
Another important characteristic of a differential equation is its order, which is the order of the highest derivative (of a dependent variable) appearing in the equation. For instance, a first-order differential equation contains only first derivatives, like both examples above.
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 approximation of the solution of a differential equation by the solution of a corresponding difference equation. See also: Time scale calculus.
Universality of mathematical description
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 second order 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, whose theory was brilliantly developed by Joseph Fourier, is governed by another second order partial differential equation, the heat equation. It turned out that many diffusion processes, while seemingly different, are described by the same equation; Black-Scholes equation in finance is for instance, related to the heat equation.
Famous differential equations
- Newton's Second Law in dynamics (mechanics)
- Hamilton's equations in classical 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
- The shallow water equations
- Picard-Lindelöf theorem on existence and uniqueness of solutions
- D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
- A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.
- W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
- E.L. Ince, Ordinary Differential Equations, Dover Publications, 1956
- lectures on differential equations MIT Open CourseWare video
- 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.
- Differential Equation Solver Java applet tool used to solve differential equations.
- Mathematical Assistant on Web online solving first order (linear and with separated variables) and second order linear differential equations (with constant coefficients), including intermediate steps in the solution, powered by Maxima
- Exact Solutions of Ordinary Differential Equations
- A bibliography of books about differential equations, from the Mathematical Association of America
- Collection of ODE and DAE models of physical systems MATLAB models
- ODE Examples with Solutions
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