Suppose and are solutions of , with p, q, and r being continuous functions of x. If and are linearly independent on an interval, is it possible to still get a Wronskian of 0 (id est and )?
-wikipedia Wronskian - Wikipedia, the free encyclopediaA common misconception is that W = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions x2 and |x|x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0.
Wronskian=0 implies that the functions are dependent if they are analytic on some intervals. The abs(x) is not analytic on the neighborhood of zero. So x^2 and |x|x are linearly dependent on any open interval not containing zero.So even though y1 & y2 seems linearly independent, they are actually not on any open interval not containing zero .