Wronskian - wikipedia Wronskian In mathematics, the Wronskian ( or Wrońskian ) is a determinant introduced by Józef Hoene - Wroński ( 1776 ) and named by Thomas Muir ( 1882, Chapter XVIII ). It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Contents 1 Definition 2 The Wronskian and linear independence 3 Application to linear differential equations 4 Generalized Wronskians 5 See also 6 Notes 7 References Definition The Wronskian of two differentiable functions f and g is W ( f, g ) = f g ′ -- g f ′. More generally, for n real - or complex - valued functions f,..., f, which are n -- 1 times differentiable on an interval I, the Wronskian W ( f,..., f ) as a function on I is defined by W ( f 1,..., f n ) ( x ) = f 1 ( x ) f 2 ( x ) ⋯ f n ( x ) f 1 ′ ( x ) f 2 ′ ( x ) ⋯ f n ′ ( x ) ⋮ ⋮ ⋱ ⋮ f 1 ( n − 1 ) ( x ) f 2 ( n − 1 ) ( x ) ⋯ f n ( n − 1 ) ( x ), x ∈ I. ( displaystyle W ( f_ ( 1 ), ldots, f_ ( n ) ) ( x ) = ( \begin ( vmatrix ) f_ ( 1 ) ( x ) &f_ ( 2 ) ( x ) & cdots &f_ ( n ) ( x ) \f_ ( 1 ) ' ( x ) &f_ ( 2 ) ' ( x ) & cdots &f_ ( n ) ' ( x ) \vdots & \vdots & ddots & \vdots \f_ ( 1 ) ^ ( ( n - 1 ) ) ( x ) &f_ ( 2 ) ^ ( ( n - 1 ) ) ( x ) & cdots &f_ ( n ) ^ ( ( n - 1 ) ) ( x ) end ( vmatrix ) ), qquad x in I. ) 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 ) th derivative, thus forming a square matrix sometimes called a fundamental matrix. When the functions f are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel 's identity, even if the functions f are not known explicitly. The Wronskian and linear independence If the functions f are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes. Thus, the Wronskian can be used to show that a set of differentiable functions is linearly independent on an interval by showing that it does not vanish identically. It may, however, vanish at isolated points. A common misconception is that W = 0 everywhere implies linear dependence, but Peano ( 1889 ) pointed out that the functions x and x x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0. There are several extra conditions which ensure that the vanishing of the Wronskian in an interval implies linear dependence. Peano ( 1889 ) observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent. ( Peano published his example twice, because the first time he published it an editor Paul Mansion, who had written a textbook incorrectly claiming that the vanishing of the Wronskian implies linear dependence, added a footnote to Peano 's paper claiming that this result is correct as long as neither function is identically zero. Peano 's second paper pointed out that this footnote was nonsense. ) Bocher ( 1901 ) gave several other conditions for the vanishing of the Wronskian to imply linear dependence ; for example, if the Wronskian of n functions is identically zero and the n Wronskians of n -- 1 of them do not all vanish at any point then the functions are linearly dependent. Wolsson ( 1989a ) gave a more general condition that together with the vanishing of the Wronskian implies linear dependence. Over fields of positive characteristic p the Wronskian may vanish even for linearly independent polynomials ; for example, the Wronskian of x and 1 is identically 0. Application to linear differential equations In general, for an n ( displaystyle n ) th order linear differential equation, if ( n − 1 ) ( displaystyle ( n - 1 ) ) solutions are known, the last one can be determined by using the Wronskian. Consider the second order differential equation in Lagrange 's notation y '' = a y ′ + b y ( displaystyle y ' ' = ay'+ by ) where a ( x ), b ( x ) ( displaystyle a ( x ), b ( x ) ) are known functions of x ( displaystyle x ) and y ( x ) ( displaystyle y ( x ) ) is the yet to be determined function. Let us call y 1, y 2 ( displaystyle y_ ( 1 ), y_ ( 2 ) ) the two solutions of the equation and form their Wronskian W ( x ) = y 1 y 2 ′ − y 2 y 1 ′ ( displaystyle W ( x ) = y_ ( 1 ) y'_ ( 2 ) - y_ ( 2 ) y'_ ( 1 ) ) Then differentiating W ( x ) ( displaystyle W ( x ) ) and using the fact that y i ( displaystyle y_ ( i ) ) obey the above differential equation shows that W ′ ( x ) = a W ( x ) ( displaystyle W ' ( x ) = aW ( x ) ) Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved : W ( x ) = e A ( x ) ( displaystyle W ( x ) = e ^ ( A ( x ) ) ) where A ′ ( x ) = a ( x ) ( displaystyle A ' ( x ) = a ( x ) ) Now suppose that we know one of the solutions, say y 2 ( displaystyle y_ ( 2 ) ). Then, by the definition of the Wronskian, y 1 ( displaystyle y_ ( 1 ) ) obeys a first order differential equation : y 1 ′ − y 2 ′ y 2 y 1 = − W ( x ) / y 2 ( displaystyle y'_ ( 1 ) - ( \frac ( y'_ ( 2 ) ) ( y_ ( 2 ) ) ) y_ ( 1 ) = - W ( x ) / y_ ( 2 ) ) and can be solved exactly ( at least in theory ). The method is easily generalized to higher order equations. Generalized Wronskians For n functions of several variables, a generalized Wronskian is a determinant of an n by n matrix with entries D ( f ) ( with 0 ≤ i < n ), where each D is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish. As in the 1 variable case the converse is not true in general : if all generalized Wronskians vanish, this does not imply that the functions are linearly dependent. However, the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth 's theorem. For more general conditions under which the converse is valid see Wolsson ( 1989b ). See also Moore matrix, analogous to the Wronskian with differentiation replaced by the Frobenius endomorphism over a finite field. Variation of parameters Notes Jump up ^ Bender, Carl M. ; Orszag, Steven A. ( 1999 ) ( 1978 ), Advanced Mathematical Methods for Scientists and Engineers : Asymptotic Methods and Perturbation Theory, New York : Springer, p. 9, ISBN 978 - 0 - 387 - 98931 - 0 References Bôcher, Maxime ( 1901 ), `` Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence '', Transactions of the American Mathematical Society, Providence, R.I. : American Mathematical Society, 2 ( 2 ) : 139 -- 149, ISSN 0002 - 9947, JFM 32.0313. 02, JSTOR 1986214, doi : 10.2307 / 1986214 Hartman, Philip ( 1964 ), Ordinary Differential Equations, New York : John Wiley & Sons, ISBN 978 - 0 - 89871 - 510 - 1, MR 0171038, Zbl 0125.32102 Hoene - Wronski, J. ( 1812 ), Réfutation de la théorie des fonctions analytiques de Lagrange, Paris Muir, Thomas ( 1882 ), A Treatise on the Theorie of Determinants., Macmillan, JFM 15.0118. 05 Peano, Giuseppe ( 1889 ), `` Sur le déterminant wronskien. '', Mathesis ( in French ), IX : 75 -- 76, 110 -- 112, JFM 21.0153. 01 Rozov, N. Kh. ( 2001 ) ( 1994 ), `` Wronskian '', in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science + Business Media B.V. / Kluwer Academic Publishers, ISBN 978 - 1 - 55608 - 010 - 4 Wolsson, Kenneth ( 1989a ), `` A condition equivalent to linear dependence for functions with vanishing Wronskian '', Linear Algebra and its Applications, 116 : 1 -- 8, ISSN 0024 - 3795, MR 989712, Zbl 0671.15005, doi : 10.1016 / 0024 - 3795 ( 89 ) 90393 - 5 Wolsson, Kenneth ( 1989b ), `` Linear dependence of a function set of m variables with vanishing generalized Wronskians '', Linear Algebra and its Applications, 117 : 73 -- 80, ISSN 0024 - 3795, MR 993032, Zbl 0724.15004, doi : 10.1016 / 0024 - 3795 ( 89 ) 90548 - X Matrix classes Explicitly constrained entries ( 0, 1 ) Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Binary Bisymmetric Block - diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Markov Metzler Monomial Moore Nonnegative Partitioned Parisi Pentadiagonal Permutation Persymmetric Polynomial Positive Quaternionic Sign Signature Skew - Hermitian Skew - symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Unitary Vandermonde Walsh Z Constant Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero Conditions on eigenvalues or eigenvectors Companion Convergent Defective Diagonalizable Hurwitz Positive - definite Stability Stieltjes Satisfying conditions on products or inverses Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Orthonormal Singular Unimodular Unipotent Totally unimodular Weighing With specific applications Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Coxeter Derogatory Distance Duplication Elimination Euclidean distance Fundamental ( linear differential equation ) Generator Gramian Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation Wedderburn X -- Y -- Z Used in statistics Bernoulli Centering Correlation Covariance Design Dispersion Doubly stochastic Fisher information Hat Precision Stochastic Transition Used in graph theory Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Skew - adjacency Tutte Used in science and engineering Cabibbo -- Kobayashi -- Maskawa Density Fundamental ( computer vision ) Fuzzy associative Gamma Gell - Mann Hamiltonian Irregular Overlap State transition Substitution Z ( chemistry ) Related terms Jordan canonical form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Quaternionic matrix Row echelon form Wronskian List of matrices Category : Matrices Retrieved from `` https://en.wikipedia.org/w/index.php?title=Wronskian&oldid=796717608 '' Categories : Ordinary differential equations Determinants Science and technology in Poland Hidden categories : CS1 French - language sources ( fr ) Talk Contents About Wikipedia Беларуская ( тарашкевіца ) ‎ Català Čeština Deutsch Español فارسی Français 한국어 Italiano עברית Nederlands 日本 語 Polski Português Русский Slovenčina Suomi Svenska Українська 中文 Edit links This page was last edited on 22 August 2017, at 17 : 20. 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Question: what does it mean if the wronskian is zero
Answer: the functions are linearly dependent