In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order . It was introduced in 1812 by the Polish mathematician Józef Wroà Âski, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.
The Wroà Âskian of two differentiable functions and is .
More generally, for real- or complex-valued functions , which are times differentiable on an interval , the Wronskian is a function on defined by
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 are solutions of a linear differential equation, the Wroà Âskian can be found explicitly using Abel's identity, even if the functions are not known explicitly. (See below.)
If the functions are linearly dependent, then so are the columns of the Wroà Âskian (since differentiation is a linear operation), and the Wroà Âskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wroà Âskian does not vanish identically. It may, however, vanish at isolated points.
A common misconception is that everywhere implies linear dependence. pointed out that the functions and have continuous derivatives and their Wroà Âskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of . There are several extra conditions which combine with vanishing of the Wronskian in an interval to imply linear dependence.
Over fields of positive characteristic the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of 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, and y is the unknown function to be found. 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 and is a constant.
Now suppose that we know one of the solutions, say . Then, by the definition of the Wroà Âskian, obeys a first order differential equation:
and can be solved exactly (at least in theory).
The method is easily generalized to higher order equations.
The relationship between the Wronskian and linear independence can also be strengthened in the context of a differential equation. If we have linearly independent functions that are all solutions of the same monic th-order homogeneous-linear ordinary differential equation (where is a linear differential operator with respect to of order less than ) on some interval , then their Wronskian is zero nowhere on . Thus, counterexamples like and (whose Wronskian is zero everywhere) or even and (whose Wronskian is zero somewhere) are ruled out; neither pair can consist of solutions to the same second-order differential equation of this type. (It's true that and are both solutions to the same third-order differential equation . But the Wronskian of the three independent solutions , , and is nowhere zero.)
For functions of several variables, a generalized Wronskian is a determinant of an by matrix with entries (with ), where each is some constant coefficient linear partial differential operator of order . 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 .
The Wroà Âskian was introduced by and given its current name by .