Thread: Wronskian and Linear Independence

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 )?

My guess is that the answer is no, but I would like someone to confirm this.

Originally Posted by Phantasma

My guess is that the answer is no, but I would like someone to confirm this.

At first I though you were correct, it will not vanish as that would show that they are in fact linearly dependent.

A 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.

-wikipedia Wronskian - Wikipedia, the free encyclopedia

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 .

Hi,
The attachment does not answer your question, and you may will know everything I say. But it does give a partial solution:

Originally Posted by Phantasma

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 )?

The problem with your example is that x|x|, not being differentiable on any interval containing 0, cannot be a solution to such an equation. And if we are restricted to an interval that does NOT contain 0, we have all positive numbers, in which or all negative numbers in which , in either case not independent of .