<P> When the functions f are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions f are not known explicitly . </P> <P> If the functions f 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 . </P> <P> A common misconception is that W = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions x and x x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0 . There are several extra conditions which ensure that the vanishing of the Wronskian in an interval implies linear dependence . Peano (1889) observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent . (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 .) Bocher (1901) gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of n functions is identically zero and the n Wronskians 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 . </P> <P> Over fields of positive characteristic p the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of x and 1 is identically 0 . </P>

What does it mean if the wronskian is zero