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 estand )?

Printable View

- Jul 14th 2013, 07:40 AMPhantasmaWronskian 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 )? - Jul 15th 2013, 10:58 AMPhantasmaRe: Wronskian and Linear Independence
My guess is that the answer is no, but I would like someone to confirm this.

- Jul 16th 2013, 01:21 PMadkinsjrRe: Wronskian and Linear Independence
At first I though you were correct, it will not vanish as that would show that they are in fact linearly dependent.

Quote:

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.

- Jul 16th 2013, 04:34 PMmopen80Re: Wronskian and Linear Independence
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 .

- Jul 16th 2013, 05:44 PMjohngRe: Wronskian and Linear Independence
Hi,

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

Attachment 28841 - Jul 19th 2013, 06:24 AMHallsofIvyRe: Wronskian and Linear Independence
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 .