Wronskians and Linear Independence

Obviously, a family of linearly dependent functions has a zero Wronskian. Many standard textbooks on differential equations (e.g., [10, Chap. 5, §5.2], [22, Chap. 1, §4], [9, Chap. 3, §7]) contain the following warning: linearly independent functions may have an identically zero Wronskian! This seems to have been pointed out for the first time by Peano [20, 21], who gave the example of the pair of functions f1(x) = x 2 and f2(x) = x|x| defined on R, which are linearly independent but whose Wronskian vanishes. Subsequently, Bôcher [1] showed that there even exist families of infinitely differentiable real functions sharing the same property. However, it is known that under some regularity assumptions, the identical vanishing of the Wronskian does imply linear dependence. The most important result in this direction is the following.