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7.5  Two real eigenvalues (Example 1:  One {\bf positive} and one {\bf negative} eigenvalue).

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

\vskip 60pt
{\includegraphics[width=100ex]{grapht-eps-converted-to.pdf}} \hfill 

\vskip 20pt
The equilibrium solution for
this system of equations is  $\left[\matrix{x_1(t) \cr x_2(t)} \right]$ = 
\vskip 20pt
This equilibrium solution is  (choose one):  
~~~
asymptotically stable  or unstable 
\vskip 20pt
Classify this critical point's type: 
\eject

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

\vskip 20pt
\hr
\vskip 20pt

At $t\rightarrow +\infty$,  $c_1  \left[\matrix{-1 \cr ~3} \right]e^{-2t}+
c_2  \left[\matrix{0 \cr 1} \right]e^{5t}$  approaches


%\vskip 60pt
\vfill
\hr
At $t\rightarrow -\infty$,  $c_1  \left[\matrix{-1 \cr ~3} \right]e^{-2t}+
c_2  \left[\matrix{0 \cr 1} \right]e^{5t}$  approaches
\vfill

\eject

7.5  Two real eigenvalues (Example 3:  Two {\bf negative} eigenvalues).

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

\vskip 60pt
{\includegraphics[width=100ex]{grapht-eps-converted-to.pdf}} \hfill 

\vskip 20pt
The equilibrium solution for
this system of equations is  $\left[\matrix{x_1(t) \cr x_2(t)} \right]$ = 
\vskip 20pt
This equilibrium solution is  (choose one):  
~~~
asymptotically stable  or unstable 
\vskip 20pt
Classify this critical point's type: 
\eject


ASSUME $c_1 \not= 0$ and $c_2 \not= 0$

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

\vskip 20pt
\hr
\vskip 20pt

At $t\rightarrow +\infty$,  $c_1  \left[\matrix{0 \cr 1} \right]e^{-5t}
+ c_2  \left[\matrix{-1 \cr ~3} \right]e^{-2t}$  approaches


%\vskip 60pt
\vfill
\hr
At $t\rightarrow -\infty$,  $c_1  \left[\matrix{0 \cr 1} \right]e^{-5t}
+ c_2  \left[\matrix{-1 \cr ~3} \right]e^{-2t}$  approaches
\vfill

\eject

7.5  Two real eigenvalues (Example 2:  Two {\bf positive}  eigenvalues).


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

\vskip 60pt
{\includegraphics[width=100ex]{grapht-eps-converted-to.pdf}} \hfill 

\vskip 20pt
The equilibrium solution for
this system of equations is  $\left[\matrix{x_1(t) \cr x_2(t)} \right]$ = 
\vskip 20pt
This equilibrium solution is  (choose one):  
~~~
asymptotically stable  or unstable 
\vskip 20pt
Classify this critical point's type: 
\eject

ASSUME $c_1 \not= 0$ and $c_2 \not= 0$

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

\vskip 20pt
\hr
\vskip 20pt

At $t\rightarrow +\infty$,  $c_1  \left[\matrix{0 \cr 1} \right]e^{5t}
+ c_2  \left[\matrix{-1 \cr ~3} \right]e^{2t}$  approaches


%\vskip 60pt
\vfill
\hr
At $t\rightarrow -\infty$,  $c_1  \left[\matrix{0 \cr 1} \right]e^{5t}
+ c_2  \left[\matrix{-1 \cr ~3} \right]e^{2t}$  approaches
\vfill



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