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Topic: Examples of differential equations

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	Ordinary differential equation - Wikipedia, the free encyclopedia
		In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable.
		Ordinary differential equations are to be distinguished from partial differential equations where y is a function of several variables, and the differential equation involves partial derivatives.
		The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century did it receive special attention.
	en.wikipedia.org /wiki/Ordinary_differential_equation (2832 words)

	Learn more about Differential equation in the online encyclopedia. (Site not responding. Last check: 2007-10-09)
		Ordinary differential equations are to be distinguished from partial differential equations where is a function of several variables, and the differential equation involves partial derivatives.
		Differential equations are used to construct mathematical models of physical phenomena such as fluid dynamics or celestial mechanics.
		Therefore, the study of differential equations is a wide field in both pure and applied mathematics.
	www.onlineencyclopedia.org /d/di/differential_equation.html (443 words)

	Examples of differential equations - Wikipedia, the free encyclopedia
		For example, if we suppose at t = 0 the extension is a unit distance (x = 1), and the particle is not moving (dx / dt = 0).
		This is a quadratic equation which we can solve.
		An exact differential equation is a first-order ordinary differential equation of implicit form
	en.wikipedia.org /wiki/Examples_of_differential_equations (695 words)

	Differential equations
		Ordinary differential equations may be classified according to the highest degree of derivative involved.
		A common though slightly informal way to express the conclusion of this example is that "the general solution of the equation is x(t) = t + C." That is, the symbol C in a solution is taken to be a constant that may take any value.
		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)

	separable differential equations examples
		The method of separation of variables is applied to the population growth in Italy and to an example of water leaking from a cylinder.
		This equation is derived using basic physics with the assumption that the sum of the kinetic and potential energy of the system remains constant.
		This differential equation is solved using the separation of variables technique.
	www-rohan.sdsu.edu /~jmahaffy/courses/f00/math122/lectures/sep_diffequations/sepdeeg.htm (1064 words)

	0.75 in (Site not responding. Last check: 2007-10-09)
		Many differential equations come from modeling some sort of natural process, such as Newton's second law, stating that the acceleration of a body will be proportional to the force acting upon it.
		Example 3 Radiocarbon Dating: A certain percentage of the carbon in the air (mostly in carbon-dioxide) is carbon-14, an apparently harmless radioactive form of carbon.
		Example 5 Classical epidemics are also modeled (for a while) by this same differential equation, since the more people who have a disease, the more they spread it to others.
	www.lehigh.edu /dlj0/courses/51ss100-51-53.html (1800 words)

	IntroDiffEq
		Differential equations can be thought of as a continuous extension of the discrete dynamical systems.
		A differential equation is any equation of some unknown function that involves some derivative of the unknown function.
		Most frequently, a differential equation is describing a growth rate, a reaction rate, or the change in some physiological state.
	www-rohan.sdsu.edu /~jmahaffy/courses/s02/math337/lectures/intro_diff_equation/intro_diffeq.html (595 words)

	Examples of differential equations (Site not responding. Last check: 2007-10-09)
		A simple Ordinary Differential Equation The simplest differential equations are ordinary, linear differential equations of the first order with constant coefficients.
		For example, if we suppose at $t = 0$ the extension is a unit distance ( $x = 1$ ), and the particle isn't moving ( $dx/dt = 0$ ).
		According to this view the form and certain vibrations--are as much our manner of perceiving that the such and such--as the colour yellow is our perception that a second, or as the action of a man walking about is our mode of.
	www.termsdefined.net /ex/examples-of-differential-equations.html (704 words)

	This course combines Differential Equations (DE) and Linear Algebra to form a powerful engine for applications (Site not responding. Last check: 2007-10-09)
		In Differential Equations we seek the solution of equations in which the unknown is a function and in which derivatives of the function appear.
		As was demonstrated in these two examples, in Differential Equations we seek solutions of equations in which the unknown is a function and in which derivatives of the function appear.
		A more interesting example is the second order DE This relatively simple DE can be used to model problems in mechanics, electricity and even in acoustics.
	www.math.fsu.edu /~fusaro/EngMath/Ch1/Introduction.html (416 words)

	Differential equations/Examples - Wikipedia
		Suppose a mass is attached to a spring, which exerts an attractive force on the mass proportional to the extension/compression of the spring and ignore any other forces (gravity, friction etc).
		For example, if we suppose at t=0 the extension is a unit distance (x=1), and the particle is not moving (dx/dt=0).
		Our new differential equation, expressing the balancing of the acceleration and the forces, is d
	nostalgia.wikipedia.org /wiki/Differential_equations/Examples (560 words)

	[No title]
		Solution of sets of linear homogeneous differential eqns -------------------------------------------------------- A feature of LIE is that the solver for systems of linear homogeneous differential equations is general and works for equations that were not necessarily obtained from point symmetry analysis or not even from any type of symmetry analysis.
		On pp 264-266 of their book "Symmetries and Differential Equations", Bluman and Kumei give the example of the third order LB syms of Burgers eqn for which they give nine syms on p 266.
		There are 3 example data files from the literature and these illustrate how substitutions from the Invariant Surface Condition and its derivatives are substituted into the target differential equation.
	archives.math.utk.edu /software/msdos/adv.diff.equations/lie/lie51.readme (3209 words)

	[No title]
		A solution to a differential equation is a relation between the variables x and y in which no derivatives appear and when we substitute it into the differential equation, that equation becomes an identity (true for all allowed values of the variables).
		A particular solution of the differential equation is given by a specific choice of the arbitrary constant: it is a solution that does not contain any arbitrary constants.
		Equations specifying the value of y and as many of its derivatives as necessary at a single chosen value of x as usually called initial conditions even if the independent variable is not time.
	www.lancs.ac.uk /users/physics/teaching/py142/lecture9.doc (1511 words)

	Differential Equations (Math 3401) - Basic Concepts - Definitions (Site not responding. Last check: 2007-10-09)
		I’ll leave the details to you to check that these are in fact solutions. Given these examples can you come up with any other solutions to the differential equation? There are in fact an infinite number of solutions to this differential equation.
		The actual solution to a differential equation is the specific solution that not only satisfies the differential equation, but also satisfies the given initial condition(s).
		From this last example we can see that once we have the general solution to a differential equation finding the actual solution is nothing more than applying the initial condition(s) and solving for the constant(s) that are in the general solution.
	tutorial.math.lamar.edu /AllBrowsers/3401/Definitions.asp (1485 words)

	Knowledge King - Examples of differential equations (Site not responding. Last check: 2007-10-09)
		The simplest differential equations are ordinary, linear differential equations of the first order with constant coefficients.
		Some elaboration is needed since is not in fact a constant, indeed it might not even be integrable.
		For example, if we suppose at the extension is a unit distance (), and the particle is not moving ().
	www.knowledgeking.net /encyclopedia/e/ex/examples_of_differential_equations.html (426 words)

	DiffEqu (Site not responding. Last check: 2007-10-09)
		The solution of the differential equation are certain functions.
		The differential equation defines the slope of at the point (x,y) of the certain curve of the function that passes through this point.
		We say that the differential equation defines the direction field of the differential equation.
	www.ies.co.jp /math/java/calc/DiffEqu/DiffEqu.html (72 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		The emphasis is on understanding and the ability to apply the theory to examples.
		Throughout the course, many examples of nonlinear differential equations will be used to illustrate basic ideas and methods.
		Differential equations are usually mathematical models of complex physical and engineering problems representing the change of processes in time.
	www.iit.edu /~duan/courses/825text (273 words)

	Introduction to Differential Equations -- from Mathematica Information Center
		To appreciate how differential equations arise from modeling real-world processes and how they can be used in real-world applications
		To understand what differential equations are, what it means to be a solution of one, and how solutions should be interpreted
		This is a sophomore level course having 3 terms of calculus as prerequisite.
	library.wolfram.com /infocenter/Courseware/4203 (329 words)

	Ordinary and Parial Differential Equations (Site not responding. Last check: 2007-10-09)
		In the first case only ordinary derivatives appear in the differential equation, and it is said to be an ordinary differential equation.
		In the second case the derivatives are partial derivatives, and the equation is called a partial differential equation.
		The potential equation, the diffusion equation, and the wave equation arise in a variety of problems in the fields of electricity and magnetism, elasticity, and fluid mechanics.
	www.cs.unc.edu /~smp/COMP205/LECTURES/DIFF/lec14/node2.html (177 words)

	Simple Differential Equations (Site not responding. Last check: 2007-10-09)
		A differential equation is any equation containing a derivative term.
		The type of differential equations that you will face in A-level Maths, can be solved using a method called separating the variables.
		Clearly this was a very simple example, but it shows the basic principles for solving simple differential equations.
	www.mathsyear2000.org /alevel/pure/purtutintsim.htm (165 words)

	Lyapunov Exponents Of Linear Stochastic Functional Differential Equations - Part Ii: Examples And Case Studies - ... (Site not responding. Last check: 2007-10-09)
		The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semiflow exists or not.
		In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate 1 (oe) of the trajectories expressed in terms of the noise variance oe.
		Mohammed, S.-E. A., and Scheutzow, M. R., Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales, Part I: The multiplicative ergodic theory, Ann.
	citeseer.ist.psu.edu /154817.html (646 words)

	Book Review of Handbook of Nonlinear Partial Differential Equations
		The stated goal of this text is impressive: to present closed-form solutions to more than 1,600 nonlinear partial differential equations.
		The equations chosen appear in multiple physical and biological sciences problems, including heat and mass transfer, hydrodynamics, control theory, and chemical engineering.
		The next two chapters address mixed derivatives and general cases of second-order differential equations, while the last three chapters are concerned with third-, fourth-, and higher-order differential equations.
	www.aip.org /tip/breview/br15.html (432 words)

	Lyapunov exponents of linear stochastic functional-differential equations. II. Examples and case studies, Salah-Eldin ...
		We give several examples and examine case studies of linear stochastic functional differential equations.
		The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semi-flow exists or not.
		These estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases.
	projecteuclid.org /getRecord?id=euclid.aop/1024404511 (474 words)

	MA11210 - DIFFERENTIAL EQUATIONS (Site not responding. Last check: 2007-10-09)
		The purpose of this module is to introduce students to the notion of mathematical modelling and to develop the technical skills for the solution of the mathematical problems that arise in applications.
		Examples will be taken from biology, economics anad physics.
		Formulation of differential equations to describe time-dependent phenomena.
	www.aber.ac.uk /modules/future/MA11210.html (199 words)

	Trafford Publishing: Mathematical Methods for Partial Differential Equations
		Mathematical Methods for Partial Differential Equations is an introduction in the use of various mathematical methods needed for solving linear partial differential equations.
		The material is suitable for a two semester course in partial differential equations for mathematicians, engineers, physicists, chemistry and science majors and is suitable for upper level college undergraduates or beginning graduate students.
		Chapter eight introduces Green's functions for ordinary differential equations and chapter nine finishes with applications of Green function techniques for solving linear partial differential equations.
	www.trafford.com /4dcgi/robots/03-0749.html (593 words)

	Examples of solution of stiff differential equations by explicit Runge-Kutta-Chebyshev method and DUMKA3
		We do not compare implicit method and dumka3 for this simple problem, because it is obvious that implicit method will be faster for this particular example, because it is very fast to inverse matrix in this particular case of one-dimensions problem.
		The situation is completely different for problems that appear after semidiscretization of large 2-3 dimensional systems of partial differential equations.
		In this example you can see that Euler method with stable(!) step size 2/a is not accurate at all, so it makes sence to compare these two methods for the same tolerance.
	www.math.tulane.edu /~amedovik/examples/examples.html (244 words)

	The practical use of Differential Equations? - Halflife2.net
		He says that the example he was given was the loading of a capacitor.
		When bored one afternoon, myself and a friend did some calculations (using differential and integral calculus, of course) to determine how high a tennis ball would bounce back up when dropped from the window of her kitchen on the 14th floor.
		Also, in musical terms, differential equations are used in modelling the sound of real instruments.
	www.halflife2.net /forums/showthread.php?t=70559 (792 words)

	Direct solution of fundamental equations (from principles of physical science) -- Encyclopædia Britannica (Site not responding. Last check: 2007-10-09)
		Insofar as the Sun and planets, with their attendant satellites, can be treated as concentrated masses moving under their mutual gravitational influences, they form a system that has not so overwhelmingly many separate units as to rule out step-by-step calculation of the motion of each.
		More results on "Direct solution of fundamental equations (from principles of physical science)" when you join.
		Features solved examples of questions in kinematics, free fall, and aspects of motion.
	www.britannica.com /eb/article-60625 (866 words)

	MATHS 374
		Catalog Description: Introduction to nth-order ordinary differential equations, equations of order one, elementary applications, linear equations with constant coefficients, nonhomogeneous equations, undetermined coefficients, variation of parameters, linear systems of equations, and the Laplace transform.
		Course Objectives: This course is an introduction to the study of differential equations.
		Linear Differential Equations: general linear equation, existence and uniqueness of solutions, linear independence, the Wronskian, general solution of a homogeneous equation, general solution of a nonhomogeneous equation.
	www.bsu.edu /math/ugsyllabi/374.htm (620 words)

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