The Wronskian of two differentiable functions and is. More generally, for real- or complex-valued functions, which are times differentiable on an interval, the Wronskian as a function on is defined by 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 th derivative, thus forming a square matrix sometimes called a fundamental matrix. When the functions are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions are not known explicitly.
The Wronskian and linear independence
If the functions 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 everywhere implies linear dependence, but pointed out that the functions and have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of. There are several extra conditions which ensure that the vanishing of the Wronskian in an interval implies linear dependence. observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent. gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of functions is identically zero and the Wronskians of of them do not all vanish at any point then the functions are linearly dependent. gave a more general condition that together with the vanishing of the Wronskian implies linear dependence. Over fields of positive characteristicp the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of xp 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. The method is easily generalized to higher order equations.
Generalized Wronskians
For functions of several variables, a generalized Wronskian is a determinant of an by matrix with entries , 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 1 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.
