# Challenging Dif EQ

• Dec 9th 2007, 10:40 PM
Auxiliary
Challenging Dif EQ
Given the following Dif EQ's (linear):

$\textbf{X}'=\left(\begin{array}{rr}-1 & c \\ 1 & -1 \end{array}\right)\textbf{X}$

Determine the values of $c$, where $c$ is a constant, for which every solution $\textbf{X}$ has the property where $\textbf{X}(t)\rightarrow \textbf{0}$ as $t\rightarrow \infty$
• Dec 10th 2007, 06:14 AM
TKHunny
Is there a determinant property that will solve this?
• Dec 10th 2007, 09:32 AM
Auxiliary
Quote:

Originally Posted by TKHunny
Is there a determinant property that will solve this?

I was thinking that maybe I have to do the Wronskian on this, but I'm not entirely sure.
• Dec 11th 2007, 05:01 PM
Auxiliary
I plugged this in Maple, and found the eigenvalues and corresponding eigenvectors in terms of c.

The two eigenvalues are:

$-1+\sqrt{c}$

$-1-\sqrt{c}$

The corresponding eigenvectors, respectively, are:

[sqrt(c),1]

[-sqrt(c),1]

The multiplicities of both are 1. Not sure where to go with this...

When c = 0, we have repeated eigenvalues ...

When c is negative, we have complex numbers... I can't imagine complex numbers yielding X(t) -> 0 with t -> infinity ....

Ahh, very challenging.