Wronskian

In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order n – 1. It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.

Definition

The Wrońskian of two differentiable functions f  and g is ${\displaystyle W(f,g)=fg'-gf'}$.

More generally, for n real- or complex-valued functions f₁, …, f_n, which are n – 1 times differentiable on an interval I, the Wronskian ${\displaystyle W(f_{1},\ldots ,f_{n})}$ is a function on ${\displaystyle x\in I}$ defined by $${\displaystyle W(f_{1},\ldots ,f_{n})(x)=\det {\begin{bmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{bmatrix}}.}$$

This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the ${\displaystyle (n-1)^{\text{th}}}$ derivative, thus forming a square matrix.

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

The Wrońskian and linear independence

If the functions f_i are linearly dependent, then so are the columns of the Wrońskian (since differentiation is a linear operation), and the Wrońskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wrońskian does not vanish identically. It may, however, vanish at isolated points.^[1]

A common misconception is that W = 0 everywhere implies linear dependence. Peano (1889) pointed out that the functions x² and |x| · x have continuous derivatives and their Wrońskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0.^{[lower-alpha 1]} There are several extra conditions which combine with vanishing of the Wronskian in an interval to imply linear dependence.

• Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wrońskian in an interval implies that they are linearly dependent.^[3]
• Bôcher (1901) gave several other conditions for the vanishing of the Wrońskian to imply linear dependence; for example, if the Wrońskian of n functions is identically zero and the n Wrońskians 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 Wrońskian.

Consider the second order differential equation in Lagrange's notation: $${\displaystyle y''=a(x)y'+b(x)y}$$ where ${\displaystyle a(x)}$, ${\displaystyle b(x)}$ are known, and y is the unknown function to be found. 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 $${\displaystyle W'(x)=a(x)W(x)}$$

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 Wrońskian, ${\displaystyle y_{1}}$ 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 Wrońskians

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).

History

The Wrońskian was introduced by Józef Hoene-Wroński (1812) and given its current name by Thomas Muir (1882, Chapter XVIII).

See also

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

Notes

1. 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 Wrońskian 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.^[2]

Citations

1. 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
2. Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation". Convergence. Mathematical Association of America. doi:10.4169/loci003642. Retrieved 2020-10-08.
3. Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation". Convergence. Mathematical Association of America. Section "On the Wronskian Determinant". doi:10.4169/loci003642. Retrieved 2020-10-08. The most famous theorem is attributed to Bocher, and states that if the Wronskian of ${\displaystyle n}$ analytic functions is zero, then the functions are linearly dependent ([B2], [BD]). [The citations 'B2' and 'BD' refer to Bôcher (1900–1901) and Bostan and Dumas (2010), respectively.]

References

• Bôcher, Maxime (1900–1901). "The Theory of Linear Dependence". Annals of Mathematics. 2 (1/4). Princeton University: 81–96. doi:10.2307/2007186. hdl:2027/hvd.hn57mn. 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, 2 (2), Providence, R.I.: American Mathematical Society: 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. 117 (8). Taylor & Francis: 722–727. arXiv:1301.6598. doi:10.4169/000298910x515785. ISSN 0002-9890. JSTOR 10.4169/000298910x515785. S2CID 9322383.
• Hartman, Philip (1964), Ordinary Differential Equations, New York: John Wiley & Sons, ISBN 978-0-89871-510-1, MR 0171038, Zbl 0125.32102
• Hoene-Wroński, Józef (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