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Topic: Differential equations

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 Differential equation - Wikipedia, the free encyclopedia
In mathematics, a differential equation is an equation in which the derivatives of a function appear as 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.
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.
en.wikipedia.org /wiki/Differential_equation   (571 words)

 Partial differential equation - Wikipedia, the free encyclopedia
Partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity, or more generally any process that is distributed in space, or distributed in space and time.
A solution of a partial differential equation is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined.
Although the issue of the existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard–Lindelöf theorem, that is far from the case for partial differential equations.
en.wikipedia.org /wiki/Partial_differential_equation   (2999 words)

 Differential Equations
On the other hand, linear differential equations are of such importance in terms of applications, theory, and solution techniques that they warrant a strong and separate emphasis.
Finally, the dif­ferential equations course is one of the few undergraduate courses where it is possible to give students a glimpse of the nature of contemporary mathematical research.
We expect students to understand the meaning of the variables and parameters in a differential equation and to be able to interpret this meaning in terms of a particular model.
www.wordtrade.com /science/mathematics/differentialequations.htm   (4849 words)

 Differential Equations Texts
The careful development of not only techniques and theory but also applications and the geometry of differential equations will provide the student with the balanced background needed to go on in his or her chosen field, be it mathematics, engineering, the sciences, or something else.
For nonlinear difference equations, a population dynamics example is thoroughly discussed (8.7) leading to the period doubling route to chaos.
The wave equation for a vibrating string is derived from physical principles in (11.4) and solved by separation of variables.
www4.ncsu.edu /eos/users/s/slc/www/BOOKS/Diff.Eq.UG.html   (1708 words)

 Differential Equations
The boundary conditions on a differential equation are the constraining values of the function at some particular value of the independent variable.
If a solution to a differential equation is found which satisfies all the boundary conditions, then it is the only solution to that equation - this is called the uniqueness theorem.
For the differential equations applicable to physical problems, it is often possible to start with a general form and force that form to fit the physical boundary conditions of the problem.
hyperphysics.phy-astr.gsu.edu /hbase/diff.html   (795 words)

 Differential equations   (Site not responding. Last check: 2007-10-22)
Response #: 2 of 2 Author: man k kwong The simplest differential equation is an equation that involves an unknown function y = f(x) and its derivatives.
A "linear differential equation" is one in which the unknowns y, y', etc. are never raised to a power more than 1, nor are they in the denominator of a fraction, nor are they inside another function such as sin.
Note that it is OK to raise x to power 2, have sin x, and multiply x^2 to y, because in the context of a differential equations, x is not considered as the unknown.
www.newton.dep.anl.gov /newton/askasci/1995/math/MATH135.HTM   (682 words)

 Differential Equations   (Site not responding. Last check: 2007-10-22)
Differential Equations are equations which contain derivatives of functions as well as the functions itself.
Slope fields are used to analyze the behavior of the solutions to a differential equation.
Some differential equations cannot be solved symbolically, and a more accurate solution may be required than that which results from a slope field analysis.
www.scit.wlv.ac.uk /university/scit/maths/calculus/modules/topics/diffeq/diffeq.htm   (161 words)

 Differential Equations
The most widely investigated differential equations are linear ones, in which the functions you are solving for, as well as their derivatives, appear only linearly.
For an ordinary differential equation, it is guaranteed that a general solution must exist, with the property that adding initial or boundary conditions simply corresponds to forcing specific choices for arbitrary constants in the solution.
Other partial differential equations can be solved only when specific initial or boundary values are given, and in the vast majority of cases no solutions can be found as exact formulas in terms of standard mathematical functions.
documents.wolfram.com /v5/TheMathematicaBook/AdvancedMathematicsInMathematica/Calculus/3.5.10.html   (1217 words)

 [No title]   (Site not responding. Last check: 2007-10-22)
Linear algebra and multivariable concepts are developed in the differential equations course as needed, so students should not feel that they are at a disadvantage if they have not completed courses in multivariable calculus and linear algebra.
Biology, in particular, is involving their students in the study of differential equations at a rapidly increasing level.
Differential equations is taught in our phsyics laboratory where each student will have access to a computer during class for discovery and demonstration.
online.redwoods.cc.ca.us /instruct/darnold/DiffEq   (417 words)

 Differential equations   (Site not responding. Last check: 2007-10-22)
Ordinary differential equations may be classified according to the highest degree of derivative involved.
of an ordinary differential equation is a function x that satisfies the equation for all values of t.
Many models specify both that a function satisfy a differential equation and that the value of the function, or the values of the derivatives of the function, take certain values for some values of the variable.
www.chass.utoronto.ca /~osborne/MathTutorial/IDE.HTM   (832 words)

 Love by the Numbers By Jordan Ellenberg
The progress of the marble is governed by a differential equation, which means, more or less, that the change in the marble's position is predictably determined by the marble's position at the moment.
The idea that marriages obey differential equations might not be so scary; after all, this only seems to say that the course of a marriage is as regular, in the long term, as weather.
Since behavior of fundamental particles is described by the (partial) differential equations of quantum mechanics, it does not take a genius and twenty years of experimental work to say that marriage, or for that matter any human activity, may eventually be described by differential equations.
www.slate.com /id/2081484   (1857 words)

 Differential equations - Hutchinson encyclopedia article about Differential equations
In a linear differential equation, the unknown function and its derivatives never appear in a power other than one.
Partial differential equations involve unknown functions of several variables, and partial derivatives do therefore appear.
This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional.
encyclopedia.farlex.com /Differential+equations   (95 words)

 Differential Equations
Differential equations provide the medium for the interaction between mathematics (especially calculus) and various branches of science and engineering.
Differential equations come in all levels of complexity and even today there is active mathematical research in differential equations.
In general, a differential equation gives a relationship between (among) a function (or functions) and its (their) derivatives (of various orders).
dept.physics.upenn.edu /courses/gladney/mathphys/java/sect2/subsubsection2_1_1_4.html   (673 words)

 34: Ordinary differential equations
Ordinary differential equations are equations to be solved in which the unknown element is a function, rather than a number, and in which the known information relates that function to its derivatives.
Few such equations admit an explicit answer, but there is a wealth of qualitative information describing the solutions and their dependence on the defining equation.
The solutions to many classic differential equations, particularly linear second-order differential equations, cannot be expressed in terms of the elementary functions but are themselves studied in 33: Special Functions.
www.math.niu.edu /~rusin/known-math/index/34-XX.html   (771 words)

 Modules for Differential Equations
Purpose: To explore the applicability of a linear differential equation as a model for the process of sprinting, and to illustrate the importance of parameters in modeling.
Prerequisites: The Spring Motion module and knowledge of the symbolic form of solutions of differential equations of the form y" + ay' + by = f(t), where f is a sine or cosine function.
Purpose: To explore the phase plane for a second-order nonlinear differential equation, specifically the standard model for damped and undamped pendulums.
www.math.duke.edu /education/ccp/resources/teach/diffeq.html   (542 words)

 An Introduction to Differential Equations
A differential equation is an equation that defines a relationship between a function and one or more derivatives of that function.
In differential equations the unknowns of concern are no longer unknown variables, but unknown functions that stand in certain mathematical relationships with their derivatives, other functions, constants, and regular variables.
And to continue the analogy yet some more, just as the equations with which you are familiar are classified into types according to their forms and solution techniques, so to with differential equations.
www.physics.ohio-state.edu /~physedu/mapletutorial/tutorials/diff_eqs/intro.html   (586 words)

 About the book
The study of differential equations is a beautiful application of the ideas and techniques of calculus to our everyday lives.
Like other texts, we begin with first-order equations, but the only analytic technique we use to find closed-form solutions is separation of variables (and, at the end of the chapter, an integrating factor or two to handle certain linear equations).
The goal of that project was to rethink the traditional, sophomore-level differential equations course.
math.bu.edu /odes/philosophy.html   (1801 words)

 Open Directory - Science: Math: Differential Equations
Differential Equations in Banach Algebras - Fuchsian Singularities of Linear Ordinary Differential Equations in Banach Algebras.
Differential Equations in Industry and Commerce - European TMR network coordinated at the Oxford Centre for Industrial and Applied Mathematics.
Numerical Methods for Partial Differential Equations - Methods such as finite differences, finite elements, fast Fourier transforms, Monte-Carlo and Lagrangian schemes are discussed in 1D to solve a variety of problems including the advection, diffusion, Black-Scholes, Burger, Korteweg-DeVries and the Schroedinger equations.
dmoz.org /Science/Math/Differential_Equations   (746 words)

 35: Partial differential equations
Like ordinary differential equations, partial differential equations are equations to be solved in which the unknown element is a function, but in PDEs the function is one of several variables, and so of course the known information relates the function and its partial derivatives with respect to the several variables.
Linear differential equations occur perhaps most frequently in applications (in settings in which a superposition principle is appropriate.) When these differential equations are first-order, they share many features with ordinary differential equations.
For example, integral techniques (solving a differential equation by computing a convolution, say) lead to integral operators (transforms on functions spaces); these and differential operators lead in turn to general pseudodifferential operators on function spaces.
www.math.niu.edu /~rusin/known-math/index/35-XX.html   (1097 words)

 MIT OpenCourseWare | Mathematics | 18.03 Differential Equations, Spring 2004 | Home   (Site not responding. Last check: 2007-10-22)
Differential Equations are the language in which the laws of nature are expressed.
Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering.
Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.
ocw.mit.edu /OcwWeb/Mathematics/18-03Spring2004/CourseHome   (104 words)

 The Math Forum - Math Library - Differential Eqtns   (Site not responding. Last check: 2007-10-22)
Elementary bifurcation theory is topic is rarely included in traditional differential equations courses, yet it is of crucial importance in many engineering applications.
An article defining and describing calculus, including differential and integral calculus, differential equations, and the development of calculus.
A computer-based course about calculus, differential equations, and matrix theory, which the instructor can use as soon as the computers are unloaded at the classroom door.
mathforum.org /library/topics/diffeq   (2333 words)

 LMS/EPSRC Short Course on Computational Differential Equations
Differential equations (DEs) are ubiquitous in science and engineering, being used for all kinds of modelling and prediction.
The course is aimed at first and second year mathematics Ph.D. students working in any area that requires computational solution of differential equations; it assumes a familiarity with numerical analysis but not a strong background in the subject.
Stochastic differential equations arise in mathematical models of physical systems which possess inherent noise and uncertainty.
www.ma.man.ac.uk /~higham/cde05   (837 words)

 Amazon.com: Differential Equations Demystified (Demystified): Books: Steven G. Krantz   (Site not responding. Last check: 2007-10-22)
Differential equations is an important subject that lies at the heart of the calculus.
A differential equation is an equation relating some function f to one or more of its derivatives.
Differential Equations Demystified is written by Steven Krantz, the author of the not-so-popular Calculus Demystified.
www.amazon.com /exec/obidos/tg/detail/-/0071440259?v=glance   (1214 words)

 DIFFERENTIAL EQUATIONS   (Site not responding. Last check: 2007-10-22)
Differential Equations, a translation of Differentsial'nye uravneniya, is devoted exclusively to differential equations and the associated integral equations.
ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral differential equations, difference equations and their applications in control theory, mathematical modelling, shell theory, informatics, and oscillation theory.
Differential Equations is abstracted and/or indexed in COMPUMATH Citation Index, INSPEC Information Services, The ISI Alerting Services, Science Citation Index Expanded.
www.maik.rssi.ru /cgi-bin/journal.pl?name=difeq&page=main   (109 words)

 Coupled differential equations
Just as we did in the unit on differential equations, we are going to solve these equations incrementally and we are going to do so by assuming that both velocity and acceleration are constant in each tiny time step.
This works because the update equation is simply taking an approximation in the definition of the derivative relationship between velocity and displacement and between acceleration and velocity.
Linear means that the differential equation is linear in terms of the dependent variable and its derivative.
othello.mech.northwestern.edu /ea3/book/diffeq3/Diffeq3.htm   (4601 words)

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