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	PlanetMath: Wronskian determinant
		For example, if we wish to determine if two solutions of a second-order differential equation are independent, we may use the Wronskian.
		As a simple illustration of this, let us consider polynomials of at most second order.
		This is version 10 of Wronskian determinant, born on 2002-02-19, modified 2006-12-31.
	planetmath.org /encyclopedia/Wronskian.html (219 words)

	The Wronskian
		Remark: Proportionality of two functions is equivalent to their linear dependence.
		Following the above discussion, we may use the Wronskian to determine the dependence or independence of two functions.
		In fact, the above discussion cannot be reproduced as is for more than two functions while the Wronskian does....
	www.sosmath.com /diffeq/second/linearind/wronskian/wronskian.html (112 words)

	Wronskian - ExampleProblems.com
		In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations.
		That is, it is the determinant of the matrix constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the n-1 derivative, thus forming a square matrix sometimes called a fundamental matrix.
		For that idea to be implemented one must be working with some formulation in which differential equations are sufficiently like vector spaces: for example in the language of vector bundles carrying a connection.
	www.exampleproblems.com /wiki/index.php/Wronskian (472 words)

	Wronskian - Wikipedia, the free encyclopedia
		In mathematics, the Wronskian is a function named after Polish mathematician Józef Hoene-Wroński, especially important in the study of differential equations.
		That is, it is the determinant of the matrix constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the n-1 derivative, thus forming a square matrix sometimes called a fundamental matrix.
		Consequently, the Wronskian determinant is zero at all points of the interval.
	en.wikipedia.org /wiki/Wronskian (490 words)

	Wronskian Matrix (Site not responding. Last check: 2007-10-28)
		The wronskian determinant, or simply the wronskian, of y_1, y_2,..., y_n is the determinant of the wronskian matrix W(y_1, y_2,..., y_n).
		Given a sequence of differential ring elements L, return the Wronskian matrix of L whose entries are elements of the universe of L. WronskianDeterminant(L) : [RngDiffElt] -> RngDiffElt, AlgMatElt
		Given a sequence of differential ring elements L, return the determinant of the Wronskian matrix of L as well as the matrix itself.
	www.math.lsu.edu /magma/text901.htm (130 words)

	Differential Equations (Math 3401) - Second Order DE's - More on the Wronskian (Site not responding. Last check: 2007-10-28)
		section we introduced the Wronskian to help us determine whether two solutions were a fundamental set of solutions. In this section we will look at another application of the Wronskian as well as an alternate method of computing the Wronskian.
		This fact is used to quickly identify linearly independent functions and functions that are liable to be linearly dependent.
		where the original Wronskian sitting in front of the exponential is absorbed into the c and the evaluation of the integral at t
	tutorial.math.lamar.edu /AllBrowsers/3401/Wronskian.asp (1443 words)

	Springer Online Reference Works
		The converse theorems are usually not true: Identical vanishing of a Wronskian on some set is not a sufficient condition for linear dependence of
		Two theorems giving sufficient conditions for linear dependence in terms of Wronskians are as follows.
		M. Böcher, "Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence" Trans.
	eom.springer.de /w/w098180.htm (325 words)

	Homogeneous/Nonhomogeneous Equations
		The usual method for determining linear independence is with a Wronskian.
		For two functions, f1(x) and f2(x), the Wronskian is defined to be:
		Find the Wronskian to determine linear independence of several functions of x.
	math.stcc.edu /DiffEq/DiffEQ41.html (470 words)

	Linear Second Order Differential Equations (Site not responding. Last check: 2007-10-28)
		Note: Since this is a 3 x 3 matrix, we must use Cramer’s Rule to find the Wronskian.
		Since the Wronskian is not equal to zero, then the functions are linearly independent.
		Since the Wronskian is equal to zero, then the functions are linearly dependent.
	www.utdallas.edu /dept/abp/higherorder.htm (191 words)

	[No title]
		# # Several Maple procedures are written for the purpose of studying the # Wronskian of two functions.
		This # denominator, though, is > pqdenom:=cos(2t)t^2+cos(2t)-tsin(2t); 2 pqdenom := cos(2 t) t + cos(2 t) - sin(2 t) t -------------------------------------------------------------------------------- > plot(pqdenom,t=-10..10); -------------------------------------------------------------------------------- > * Maple V Graphics # The Wronskian** becomes zero, but only when the denominator in p # and q is zero so that p and q are not continuous.
		The DE really only # makes sense in certain relatively short intervals...the coefficients # (forces, frictions) become infinite at the endpoints of those intervals.
	calclab.math.tamu.edu /docs/math308/systems/wronskian.txt (396 words)

	wronksian
		When you use the Wronskian to check for linear independence, what you are actually doing is checking if the two solutions are multiples of each other.
		If the Wronskian yields anything else, the solutions are linearly independant and they are not multiples of each other.
		The Wronskian, in that case, evaluated at x= 0 or whatever number, is simply the coefficient matrix.
	www.physicsforums.com /showthread.php?p=1154762#post1154762 (581 words)

	Amazon.com: Wronskian (Site not responding. Last check: 2007-10-28)
		This determinant is called the Wronskian of the functions fl, f2_.,...
		a well-known property of a Wronskian of second-order linear differential equations...
		determinant W is called the Wronskian of the pair yl, Y2.
	www.amazon.com /s?ie=UTF8&keywords=Wronskian&index=blended&page=1 (1176 words)

	Wronskian Proof
		I'm working on a proof by induction of the Wronskian and need a little boost to get going.
		In general, a set of functions will be linearly independent IFF the Wronskian is not identically zero.
		If b is not 0, then g(x)= (a/b)f(x) and the Wronskian if f(x)g'(x)- f'(x)g(x)= f(x)(b/a)f'(x)- f'(x)(b/a)f(x)= (b/a)(f(x)f'(x)- f'(x)f(x))= 0 for all x.
	www.physicsforums.com /showthread.php?t=74276 (632 words)

	374Lab1.nb (Site not responding. Last check: 2007-10-28)
		Check that the six "different" functions found are linearly independent and that the given general solution is actually a solution of the differential equation!
		The following set of commands saves the six functions in memory as FS, sets up the n x n matrix with rows corresponding to the functions and the first n-1 derivatives of each function, and defines the Wronskian to be the determinant of this matrix.
		The Wronskian is the determinant of this n x n matrix:
	www.bsu.edu /web/mkarls/374Lab1 (693 words)

	DC MetaData pour: On the Wronskian combinants of binary forms (Site not responding. Last check: 2007-10-28)
		Résumé: For generic binary forms $A_1,...,A_r$ of order $d$ we construct a class of combinants $C = \{\C_q: 0 \le q \le r, q \neq 1\}$, to be called the Wronskian combinants of the $A_i$.
		We show that the collection $C$ gives a projective embedding of the Grassmannian $G(r,S_d)$, and as a corollary, any other combinant admits a formula as an iterated transvectant in the $C$.
		These collections are the ones such that an associated algebraic differential equation has the maximal number of linearly independent polynomial solutions.
	www-math.univ-paris13.fr /prepub/pp2005/pp2005-19.html (158 words)

	wronskian_2_7.nb (Site not responding. Last check: 2007-10-28)
		We need a method to quickly verify the linear independence of a given set of such equations.
		The Wronskian determinant is used as a criterion for linear independence.
		are linearly independent if and only if their Wronskian W ≡ 0.
	www.ireap.umd.edu /~nmoody/Math/wronskian.html (171 words)

	Differential Equations I
		0), then this Wronskian is not 0 and we have a fundamental set.
		Selecting either of the original solutions as a particular solution, I now have all the ingredients I need for the general solution of the equation.
		Let us notice first that the leading coefficient of the equation is 1 (so never zero) and all coefficients are nice and continuous; solutions exist everywhere and if the Wronskian is different from 0 at any point, it's different from 0 at all points.
	www.math.fau.edu /schonbek/htmdocs/desu01sgs.html (908 words)

	[No title] (Site not responding. Last check: 2007-10-28)
		We have, together with (1), a system of n homogeneous equations in EMBED Equation.3: (2) EMBED Equation.3 By hypothesis, the Wronskian of a set of n functions EMBED Equation.3 is nonzero at a certain point xo, i.e.
		(b) The Wronskian is EMBED Equation.3 Example 7.2 Show that the functions EMBED Equation.3 are linear dependent for all values of x.
		Solution 7.2 The Wronskian of the functions EMBED Equation.3 is EMBED Equation.3 for all values of x Therefore the functions EMBED Equation.3 are linear dependent for all values of x.
	www.physics.hku.hk /~phys1315/Lecture_Notes/ma07np.doc (1169 words)

	Mathematics Calculus Homework Help
		They must be shown linear indenpendent by using Wronskian.
		Proof of a property of Wronskian for a given differential equation.
		What are the antiderivatives of: COS^2 theta and SIN^2 theta and how did you find these antiderivatives.
	www.brainmass.com /homework-help/math/calculus/pg224 (527 words)

	Citebase - Notes on solutions in Wronskian form to soliton equations: KdV-type
		We take the KdV equation and the Toda lattice to serve as two examples for solutions in Wronskian form and Casoratian form, respectively.
		We also discuss Wronskian solutions for the KP equation.
		Finally, we formulate the Wronskian technique as four steps.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:nlin/0603008 (199 words)

	On the solutions of Bessel's differential equation
		In Section 2 we studied the linear second order differential equation and found that the (Wronskian)
		The Wronskian for the v's then transforms into the statement
		From the general results in Section 2 we know that the Wronskian (169) must be valid, and hence the O(x)-terms in (171) must cancel.
	www.nbi.dk /~polesen/borel/node17.html (202 words)

	Higher Order Linear Ordinary Differential Equations and Solutions
		In other words, if none of these functions can be expressed in terms (by linear combination) of others, these functions are linearly independent on the interval
		The Wronskian of this set of function is
		If the Wronskian is zero, this set of functions is linearly dependent.
	www.efunda.com /math/ode/linearode_terms.cfm (257 words)

	[No title]
		%In response to questions in class, here %is how you could use Matlab to check that five functions are %linearly independent by taking the Wronskian.
		>> syms t >> f = [cos(t) sin(t) cos(2t) sin(t)^2 cos(t)^2] f = [ cos(t), sin(t), cos(2t), sin(t)^2, cos(t)^2] % I check first if somehow matlab will do this for me, no luck >> help wronskian wronskian.m not found.
		% They are linearly dependent since cos(2t) = cos(t)^2 - sin(t)^2 % So we should get zero for the Wronskian*.
	www.math.umd.edu /~hck/Wronskian.txt (250 words)

	374Wronskian.nb (Site not responding. Last check: 2007-10-28)
		Make an n x n matrix of functions whose ith row contains the (i-1)st derivatives of {f1, f2,...
		To compute the Wronskian, which is a function of x, evaluate W at x:
		Since the Wronskian is never zero, we conclude that the functions {E^x,Sin[x], Cos[x]} are linearly independent on any real interval!
	www.bsu.edu /web/mkarls/374Wronskian (89 words)

	Outline of Lesson 27
		The determinant of the Wronskian matrix is called the
		is nonzero, the Wronskian does not vanish on the real line.
		The determinants of the matrices in which a column of the Wronskian matrix is replaced with
	www.runet.edu /~thompson/437/DouglasMeade/linearequations.mw (3163 words)

	unit19.html
		To verify that this is a fundamental set of solutions, observe that both functions are solutions
		denote the determinant of the Wronskian matrix of the fundamental set and
		The determinants of the matrices in which a column of the Wronskian matrix is replaced with
	www.adeptscience.co.uk /products/mathsim/maple/powertools/des/unit19.html (2752 words)

	DC MetaData for: Wronskian solutions of the constrained KP hierarchy (Site not responding. Last check: 2007-10-28)
		DC MetaData for: Wronskian solutions of the constrained KP hierarchy
		A large class of solutions - among them solitons - can be represented
		by Wronskian determinants of functions satisfying a set of linear equa-
	math-www.uni-paderborn.de /preprints/preprints_data/Woevel/Strampp_Wronskians_96_05_06_metadata.html (60 words)

	5y - Webled.com
		[ I am helping my daughter to get the Wronskian from the ODE: y" - 2ty' ]...
		The Wronskian that you and I find is using Abel's theorem.
		[ Post subject: The Wronskian of y"-2ty' ]...
	www.webled.com /5y.htm (762 words)

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