Chapter 6.3, Problem 15P

Find constant $3\times 1$ vectors ${\text{u}}_{\text{1}}$ and ${\text{u}}_{\text{2}}$ such that the solution of the initial value problem

$\text{x}\text{'}=\left(\begin{array}{ccc}2& -4& -3\\ 3& -5& -3\\ -2& 2& 1\end{array}\right)\text{x,}\text{x}\left(0\right)={\text{x}}_0$

tends to ${\left(\begin{array}{ccc}0& 0& 0\end{array}\right)}^T$ as $t\to \infty$ for any ${\text{x}}_0\in S$, where

$S=\left\{\text{u}:\text{u}=a_{\text{1}}{\text{u}}_1+a_2{\text{u}}_{\text{2}},-\infty <a_1,a_2<\infty \right\}$.

In ${\text{R}}^3$, plot solutions to the initial value problem for several different choices of ${\text{x}}_0\in S$ overlaying the trajectories on a graph of the plane determined by ${\text{u}}_{\text{1}}$ and ${\text{u}}_{\text{2}}$. Describe the behavior of solutions as $t\to \infty$ if ${\text{x}}_{\text{0}}\notin S$.