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

The Wrońskian of two differentiable functions *f* and *g* is .

More generally, for *n* real- or complex-valued functions *f*_{1}, …, *f _{n}*, which are

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 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

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.

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

- 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 Wrońskian implies linear dependence.

Over fields of positive characteristic *p* the Wrońskian may vanish even for linearly independent polynomials; for example, the Wrońskian 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 Wrońskian.

Consider the second order differential equation in Lagrange's notation:

where , are known, and y is the unknown function to be found. Let us call the two solutions of the equation and form their Wrońskian

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

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

where and is a constant.

Now suppose that we know one of the solutions, say . Then, by the definition of the Wrońskian, 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 Wrońskian** is a determinant of an *n* by *n* matrix with entries *D _{i}*(

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