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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.

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*_{1}, . . . , *f _{n}*, which are

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

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.

A common misconception is that *W* = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions *x*^{2} and |*x*|* · x* have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0.

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.

In general, for an th order linear differential equation, if solutions are known, the last one can be determined by using the Wronskian.

Consider the second order differential equation in Lagrange's notation

where are known. Let us call the two solutions of the equation and form their Wronskian

Then differentiating and using the fact that obey the above differential equation shows that

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

where

Now suppose that we know one of the solutions, say . Then, by the definition of the Wronskian, obeys a first order differential equation:

and can be solved exactly (at least in theory).

The method is easily generalized to higher order equations.

For *n* functions of several variables, a **generalized Wronskian** is a determinant of an *n* by *n* matrix with entries *D _{i}*(

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

**^**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.^{[2]}

**^**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**^**Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation".*Convergence*. Mathematical Association of America. doi:10.4169/loci003642. Retrieved 2020-10-08.**^**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

*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.]*

- 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