<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> 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>
what does it mean if the wronskian is zero
what does it mean if the wronskian is zero
the functions are linearly dependent
the functions are linearly dependent