Chapter 6.4, Problem 9P

(a) Find constant $3\times 1$ vectors $\text{a}$ and $\text{b}$ such that the solution of the initial value problem

$\text{x}\text{'}=\left(\begin{array}{ccc}3/4& 29/4& -11/2\\ -3/4& 3/4& -5/2\\ 5/4& 11/4& -5/2\end{array}\right)\text{x, }\text{x}\left(0\right)={\text{x}}_0\in S$

is a closed curve lying entirely in $S$, where

$S=\left\{\text{u:u}=a_1\text{a+}{a}_2\text{b,-}\infty \text{<}{a}_1,a_2<\infty \right\}$

(b) In ${\text{R}}^{\text{3}}$, plot the solutions to the initial value problem for several different choices of ${\text{x}}_{\text{0}}\in S$ overlaying the trajectories on a graph of the plane determined by $\text{a}$ and $\text{b}$.

(c) In ${\text{R}}^3$, plot the solution to the initial value problem for a choice of ${\text{x}}_{\text{0}}\notin S$. Describe the long-time behavior of the trajectory.