Wronskian

Differential equations
Navier–Stokes differential equations used to simulate airflow around an obstruction
show Scope Natural sciences Engineering Astronomy Physics Chemistry Biology Geology Applied mathematics Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics
Classification
show Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear By variable type Dependent and independent variables Autonomous Complex Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation
show Relation to processes Difference (discrete analogue) Stochastic Stochastic partial Delay
Solution
show Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem
show General topics Wronskian Phase portrait Phase space Lyapunov / Asymptotic / Exponential stability Rate of convergence Series / Integral solutions Numerical integration Dirac delta function
show Solution methods Inspection Method of characteristics Euler Exponential response formula Finite difference (Crank–Nicolson) Finite element Infinite element Finite volume Galerkin Petrov–Galerkin Integrating factor Integral transforms Perturbation theory Runge–Kutta Separation of variables Undetermined coefficients Variation of parameters
show People Isaac Newton Gottfried Leibniz Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Kutta
v t

In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by Józef Hoene-Wroński (1812) 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.

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 f₁, . . . , f_n, which are n – 1 times differentiable on an interval I, the Wronskian W(f₁, . . . , f_n) as a function on I is defined by

${\displaystyle W(f_{1},\ldots ,f_{n})(x)={\begin{vmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{vmatrix}},\qquad x\in I.}$

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.

When the functions f_i are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions f_i are not known explicitly.

The Wronskian and linear independence[]

If the functions f_i 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.^[1]

A common misconception is that W = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions x² and |x| · x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0.^[a] There are several extra conditions that ensure that the vanishing of the Wronskian in an interval implies linear dependence. Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent.^[3] Bôcher (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^p and 1 is identically 0.

Application to linear differential equations[]

In general, for an ${\displaystyle n}$th order linear differential equation, if ${\displaystyle (n-1)}$ solutions are known, the last one can be determined by using the Wronskian.

Consider the second order differential equation in Lagrange's notation

${\displaystyle y''=a(x)y'+b(x)y}$

where ${\displaystyle a(x),b(x)}$ are known. Let us call ${\displaystyle y_{1},y_{2}}$ the two solutions of the equation and form their Wronskian

${\displaystyle W(x)=y_{1}y'_{2}-y_{2}y'_{1}}$

Then differentiating ${\displaystyle W(x)}$ and using the fact that ${\displaystyle y_{i}}$ obey the above differential equation shows that

${\displaystyle W'(x)=aW(x)}$

Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved:

${\displaystyle W(x)=C~e^{A(x)}}$

where ${\displaystyle A'(x)=a(x)}$ and ${\displaystyle C}$ is a constant.

Now suppose that we know one of the solutions, say ${\displaystyle y_{2}}$. Then, by the definition of the Wronskian, ${\displaystyle y_{1}}$ obeys a first order differential equation:

${\displaystyle y'_{1}-{\frac {y'_{2}}{y_{2}}}y_{1}=-W(x)/y_{2}}$

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_i(f_j) (with 0 ≤ i < n), where each D_i 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 single 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[]

• Variation of parameters
• Moore matrix, analogous to the Wronskian with differentiation replaced by the Frobenius endomorphism over a finite field.
• Alternant matrix
• Vandermonde matrix

Notes[]

Citations[]

References[]

• Bôcher, Maxime (1900–1901). "The Theory of Linear Dependence". Annals of Mathematics. Princeton University. 2 (1/4): 81–96. doi:10.2307/2007186. ISSN 0003-486X. JSTOR 2007186.
• Bôcher, Maxime (1901), "Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence" (PDF), Transactions of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 2 (2): 139–149, doi:10.2307/1986214, ISSN 0002-9947, JFM 32.0313.02, JSTOR 1986214
• Bostan, Alin; Dumas, Philippe (2010). "Wronskians and Linear Independence". American Mathematical Monthly. Taylor & Francis. 117 (8): 722–727. arXiv:1301.6598. doi:10.4169/000298910x515785. ISSN 0002-9890. JSTOR 10.4169/000298910x515785.
• 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", Encyclopedia of Mathematics, EMS Press
• Wolsson, Kenneth (1989a), "A condition equivalent to linear dependence for functions with vanishing Wronskian", Linear Algebra and its Applications, 116: 1–8, doi:10.1016/0024-3795(89)90393-5, ISSN 0024-3795, MR 0989712, Zbl 0671.15005
• Wolsson, Kenneth (1989b), "Linear dependence of a function set of m variables with vanishing generalized Wronskians", Linear Algebra and its Applications, 117: 73–80, doi:10.1016/0024-3795(89)90548-X, ISSN 0024-3795, MR 0993032, Zbl 0724.15004