# Wronskian and Linear Independence

• Jul 14th 2013, 07:40 AM
Phantasma
Wronskian and Linear Independence
Suppose \$\displaystyle y_1\$ and \$\displaystyle y_2\$ are solutions of \$\displaystyle y''+p(x)y'+q(x)y=r(x)\$, with p, q, and r being continuous functions of x. If \$\displaystyle y_1\$ and \$\displaystyle y_2\$ are linearly independent on an interval, is it possible to still get a Wronskian of 0 (id est \$\displaystyle x^2\$ and \$\displaystyle x|x|\$)?
• Jul 15th 2013, 10:58 AM
Phantasma
Re: 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 PM
Re: Wronskian and Linear Independence
Quote:

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.

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.
-wikipedia Wronskian - Wikipedia, the free encyclopedia
• Jul 16th 2013, 04:34 PM
mopen80
Re: 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 PM
johng
Re: 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 AM
HallsofIvy
Re: Wronskian and Linear Independence
Quote:

Originally Posted by Phantasma
Suppose \$\displaystyle y_1\$ and \$\displaystyle y_2\$ are solutions of \$\displaystyle y''+p(x)y'+q(x)y=r(x)\$, with p, q, and r being continuous functions of x. If \$\displaystyle y_1\$ and \$\displaystyle y_2\$ are linearly independent on an interval, is it possible to still get a Wronskian of 0 (id est \$\displaystyle x^2\$ and \$\displaystyle x|x|\$)?

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 \$\displaystyle x|x|= x^2\$ or all negative numbers in which \$\displaystyle x|x|= -x^2\$, in either case not independent of \$\displaystyle x^2\$.