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Topic: Linear differential equation

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	DIFFERENTIAL EQUATION - Online Information article about DIFFERENTIAL EQUATION
		The relation ex-pressing the equation of the envelope is called a singular solution of the differential equation, meaning an isolated solution, as not being one of a family of curves depending upon an arbitrary parameter.
		Of such equations a simple case is expressed by the pair dt =ax+by+c, dt =a'x+b'y+c', wherein the coefficients a, b, c, a', b', c', are constants.
		Differential equations arise in the expression of the relations between quantities by the elimination of details, either unknown or regarded as unessential to the formulation of the relations in question.
	encyclopedia.jrank.org /DEM_DIO/DIFFERENTIAL_EQUATION.html (10266 words)

	Differential equation Summary
		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.
	www.bookrags.com /Differential_equation (2081 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.
		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.
		Linear and nonlinear: A differential equation is said to be linear if each term in the equation has only one order of derivative, e.g., no term has both y and the derivative of y with respect to time.
	hyperphysics.phy-astr.gsu.edu /hbase/diff.html (795 words)

	PlanetMath: differential equation
		A differential equation is an equation involving an unknown function of one or more variables, its derivatives and the independent variables.
		If a differential equation is satisfied by a function which identically vanishes (i.e.
		This is version 8 of differential equation, born on 2002-05-30, modified 2005-05-09.
	planetmath.org /encyclopedia/DifferentialEquation.html (432 words)

	Linear differential equation - Wikipedia, the free encyclopedia
		where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function.
		If y is assumed to be a function of only one variable, one speaks about an ordinary differential equation, else the derivatives and their coefficients must be understood as (contracted) vectors, matrices or tensors of higher rank, and we have a (linear) partial differential equation.
		(x) by either the method of undetermined coefficients or the method of variation of parameters; the general solution to the linear differential equation is the sum of the general solution of the related homogeneous equation and the particular solution.
	en.wikipedia.org /wiki/Linear_differential_equation (827 words)

	PlanetMath: second order linear differential equation with constant coefficients
		The explicit solution is easily found using the characteristic equation method.
		Remark that the roots of (2) are the eigenvalues of the Jacobian matrix of (5).
		This is version 5 of second order linear differential equation with constant coefficients, born on 2003-02-02, modified 2006-09-16.
	planetmath.org /encyclopedia/Sink2.html (294 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 /v4/MainBook/3.5.10.html (1184 words)

	Modules for Differential Equations
		Second-Order Linear Homogeneous Differential Equations with Constant Coefficients
		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.
	www.math.duke.edu /education/ccp/resources/teach/diffeq.html (542 words)

	Ordinary differential equation (Site not responding. Last check: 2007-10-10)
		Ordinary differential equations are to be distinguished from partial differential equations where there are several independent variables involving 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.
		Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and if so, what are the characteristic properties of this function.
	www.tocatch.info /en/ODE.htm (1618 words)

	What is nonlinear dynamics?
		A linear dynamical system is one in which the dynamic rule is linearly proportional to the system variables.
		Linear systems can be analyzed by breaking the problem into pieces and then adding these pieces together to build a complete solution.
		To solve this linear differential equation we must find some function x(t) with the following property: the second derivative of x (with respect to the independent variable t) is equal to -x.
	cnls.lanl.gov /People/nbt/Book/node4.html (1508 words)

	solving second order linear difference equation
		Note that we have transformed the original equation into the form discussed at the end of the introduction to differential equations.
		We have now found an equation that yields y as a function of t, and it would be nice to say that we are finished.
		Finally, this is the general solution of the homogeneous linear differential equation.
	www.softmath.com /tutorials2/solving-second-order-linear-difference-equation.html (1154 words)

	Ordinary differential equation - Wikipedia, the free encyclopedia
		In the case where the equation is linear, it can be solved by analytical methods.
		A particular solution to the inhomogeneous equation can be found by the method of undetermined coefficients or the method of variation of parameters.
		This equation F(z) = 0, is the "characteristic" equation considered later by Monge and Cauchy.
	en.wikipedia.org /wiki/Ordinary_differential_equation (1655 words)

	First-Order Nonhomogeneous Linear Differential Equations
		But, this equation is a homogeneous linear differential equation, and we know how to solve it (yet again an an instance of reducing a problem to one that we already know how to solve).
		This is the general form of the integrating factor for all first-order nonhomogeneous linear differential equations.
		Solving initial-value problems of nonhomogeneous linear differential equations is directly analogous to solving them in the homogeneous case.
	www.physics.ohio-state.edu /~physedu/mapletutorial/tutorials/diff_eqs/non_homo.htm (984 words)

	Homogeneous/Nonhomogeneous Equations
		Here it refers to the fact that the linear equation is set to 0.
		To be linearly independent means that none of the equations can be written as a linear combination of the others.
		The general method for solving non-homogeneous differential equations is to solve the homogeneous case first and then solve for the particular solution that depends on g(x).
	math.stcc.edu /DiffEq/DiffEQ41.html (470 words)

	Springer Online Reference Works
		The general solution of (2) is a linear combination, with arbitrary constant coefficients, of the fundamental system of solutions.
		The inhomogeneous equation (1) can be integrated by the method of variation of constants.
		The general solution of (6) is a linear combination, with arbitrary constant coefficients, of the solutions that form the fundamental system.
	eom.springer.de /l/l059370.htm (339 words)

	Linear, First Order
		In the world of differential equations, the function y and its derivatives are treated as variables, while expressions in x are often manipulated as though they were constants.
		When the right side is zero we have a homogeneous linear equation, which provides the "homogeneous solutions".
		Linear equations sound innocent enough, but they can be very complicated, and we're not going to analyze them all on this website.
	www.mathreference.com /ca-ode,leq.html (389 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Linear Differential Equation A nonhomogeneous nth-order linear differential equation has the form EMBED Equation.3 (7.1) where EMBED Equation.3 and the coefficients EMBED Equation.3 depend solely on the variable x.
		Linear Homogeneous Second-order Differential Equations In particular, for a homogeneous second-order linear equation of the form EMBED Equation.3 (7.5) If EMBED Equation.3 and EMBED Equation.3 are two independent solutions of the equation, then EMBED Equation.3 (7.6) is the general solution of the equation Eq.(7.5).
		Then the linear equations has unique solutions for EMBED Equation.3 and EMBED Equation.3 which are given by (3) EMBED Equation.3 and (4) EMBED Equation.3 Theorem 7.4 Suppose EMBED Equation.3 is a non-zero solution of the equation Eq.(7.5), then EMBED Equation.3 (7.7) is a particular solution to the equation.
	www.physics.hku.hk /~phys1315/Lecture_Notes/ma07np.doc (1169 words)

	Second Order Linear Equations
		The theory of linear differential equations is closely related to the theory of systems of linear equations in linear algebra.
		This is a linear transformation in the sense of linear algebra, but the ``vectors'' it operates on are functions.
		For a set of two functions, linear independence is easy to check: they are linearly independent unless one is a constant multiple of the other.
	www.math.ubc.ca /~israel/m215/lin2o/lin2o.html (653 words)

	Linear Homogeneous Ordinary Differential Equations with Constant Coefficients
		Linear Homogeneous Ordinary Differential Equations with Constant Coefficients
		A linear homogeneous ordinary differential equation with constant coefficients has the general form of
		This equation implies that the solution is a function whose derivatives keep the same form as the function itself and do not explicitly contain the independent variable
	www.efunda.com /math/ode/linearode_consthomo.cfm (170 words)

	Integrating Factors
		However, this simple method of solution works only because the original differential equation was homogeneous, i.e., the right hand side of the original equation was zero. The solution is slightly more complicated if the right hand side is not zero. Consider an ordinary first-order differential equation of the form
		A simpler way of expressing the general solution is to begin by dividing through the original differential equation (1) by a(x), so that the leading coefficient equals 1. Then we have new functions b(x) and c(x) (equal to b/a and c/a in terms of the original functions) such that
		Therefore the differential equation is equivalent to the algebraic condition that f(x,y) is invariant, i.e., the solution is of the form
	www.mathpages.com /home/kmath220/kmath220.htm (775 words)

	AERADE
		The merit of the method lies in the fact that the application of the transform reduces such an equation to a simple algebraic one from which the required solution is found by reference to tables of transforms.
		Some of the methods are applicable to a set of n first-order differential equations with an appropriate set of n initial conditions specified at a single value of the independent variable.
		It is also shown that a differential equation of higher order than one, provided it is possible to isolate the highest derivative of the dependent variable, can be treated using the same methods.
	aerade.cranfield.ac.uk /subject-listing/esdu/ES90.html (1138 words)

	CT Linear Systems and Differential Equations
		It is easy to show that these equations define a system that is linear and time invariant.
		Where this equation is referred to as the characteristic equation of the system.
		Before deriving the convolution procedure, we show that a system's impulse response is easily derived from its linear, differential equation (LDE).
	cnx.org /content/m10855/latest (547 words)

	kentox: Green's functions!
		In a linear differential equation, the notion of superposition applies -- namely, that if you have a bunch of different solutions, then the sum of them is also a solution.
		If a differential equation is linear, and you can obtain a continuum of solutions to it, then you should be able to use integrals to get an extra solution to the problems.
		Partial differential equations are notorious for requiring an entire function to define their initial condition.
	kentox.livejournal.com /281511.html (1809 words)

	first-order linear differential equation calculator
		The poles of the Laplace transform fill the same role as the eigenvalues of a linear system (of which they are a generalization).
		This section is a standard presentation of Laplace transforms appliedto first-order equations with discontinuous terms.
		This section is a relatively standard discussion of the Laplacetransform method applied to second-order linear equations.
	www.softmath.com /tutorials2/first-order-linear-differential-equation-calculator.html (1043 words)

	Decomposing a 4'th order linear differential equation as a symmetric product. (Site not responding. Last check: 2007-10-10)
		The case that is handled is when the equation has solutions which are products of solutions of second order equations.
		From a mathematical point of view this may not seem interesting, however, there is one intriguing feature about these formulas: It turns out that, starting with an equation of order 4, to determine if it is a symmetric product of 2nd order equations one has to solve an equation of order 3, i.e.
		So we can expect that the equation to be solved to find {L1,L2} has a 3-dimensional vector space of solutions and thus will be a linear ode with order 3.
	www.math.fsu.edu /~hoeij/papers/symprod/comments.html (297 words)

	Notes on 2nd order d.e.'s
		``Linear with constant coefficients'' means that each term in the equation is a constant times y or a derivative of y.
		This equation is called the characteristic equation (associated to the differential equation).
		The method is to write down what the initial conditions mean in terms of the general solution, and then to solve for p and q.
	www.math.sunysb.edu /~blaine/mat132/denotes/denotes.html (627 words)

	Homogeneous and Nonhomogeneous Differential Equations (Site not responding. Last check: 2007-10-10)
		A homogeneous linear differential equation has the property that every term in the equation contains the dependent variable or one of its derivatives.
		A linear differential equation which does not satisfy this definition is nonhomogeneous.
		An example of a homogeneous differential equation is provided by the conduction equation in a constant-property material, not subject to volumetric heating or cooling
	www.coe.uncc.edu /~rkeanini/funht/not3h/node3.html (178 words)

	Systems of first-order linear differential equations (Site not responding. Last check: 2007-10-10)
		I give only one example, which shows how the trigonometric functions may emerge in the solution of a system of two simultaneous linear equations, which, as we saw above, is equivalent to a second-order equation.
		You are asked, in an exercise, to verify this solution by using the technique discussed at the start of this section to convert the two-equation system to a single second-order linear differential equation.
		Exponentials, the trigonometric functions, and complex numbers (which arise as roots of the characteristic equation in the technique of the previous section), are closely related.
	www.chass.utoronto.ca /~osborne/MathTutorial/SIM.HTM (469 words)

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