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7.5  Two real eigenvectors: Graph several trajectories for the following systems of equations:



 Example 1: ~ Given that the solution to
 to ${\bf x}' =
  \left[\matrix{-2 & 0 \cr 21 & 5} \right]{\bf x}$ ~is~~ ${\bf x} = c_1  \left[\matrix{-1 \cr ~3} \right]e^{-2t}+
c_2  \left[\matrix{0 \cr 1} \right]e^{5t}$




{\includegraphics[width=40ex]{grapht-eps-converted-to.pdf}} \hfill 

\vskip -2in

\rightline{
The equilibrium solution for \hskip 1.1in~}
\rightline{this system of equations is  \hskip 1in~}
\vskip 20pt
\rightline{This equilibrium solution is  (choose one):  \hskip 0.3in~}
\vskip 8pt
\rightline{asymptotically stable  or unstable \hskip .4in~}

\vskip 20pt
\rightline{Classify this critical point's type:  \hskip .85in~}

\vskip 0.8in
\hr
 Example 2: ~Given that the solution to
 ${\bf x}' =
  \left[\matrix{-2 &~ 0 \cr -9 & -5} \right]{\bf x}$ ~is~~ ${\bf x} = c_1  \left[\matrix{0 \cr 1} \right]e^{-5t}
+ c_2  \left[\matrix{-1 \cr ~3} \right]e^{-2t}$

{\includegraphics[width=40ex]{grapht-eps-converted-to.pdf}} \hfill 

\vskip -2in

\rightline{
The equilibrium solution for \hskip 1.1in~}
\rightline{this system of equations is  \hskip 1in~}
\vskip 25pt
\rightline{This equilibrium solution is  (choose one):  \hskip 0.3in~}
\vskip 10pt
\rightline{asymptotically stable  or unstable \hskip .4in~}

\vskip 20pt
\rightline{Classify this critical point's type:  \hskip .85in~}

\vskip 0.7in
\hr

 Example 3: ~ Given that the solution to ${\bf x}' =
  \left[\matrix{2 & 0 \cr 9 & 5} \right]{\bf x}$ ~is~~ ${\bf x} = c_1  \left[\matrix{0 \cr 1} \right]e^{5t}
+ c_2  \left[\matrix{-1 \cr ~3} \right]e^{2t}$

{\includegraphics[width=40ex]{grapht-eps-converted-to.pdf}} \hfill 

\vskip -2in

\rightline{
The equilibrium solution for \hskip 1.1in~}
\rightline{this system of equations is  \hskip 1in~}
\vskip 25pt
\rightline{This equilibrium solution is  (choose one):  \hskip 0.3in~}
\vskip 10pt
\rightline{asymptotically stable  or unstable \hskip .4in~}

\vskip 20pt
\rightline{Classify this critical point's type:  \hskip .85in~}


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