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Quiz 6 Form A, 
Dec 1, 2017
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[10]~  1a.) Find the general solution to the following differential equation system.
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\centerline{${\bf X'} = \left[\matrix{
1 & -1 \cr 6 &-4
 }\right]{\bf X}$}



$\left[\matrix{
1-r & -1 \cr 6 &-4-r
 }\right]{\bf X} = (1-r)(-4-r) + 6 = r^2 + 3r -4 + 6 =r^2 + 3r + 2 = (r+1)(r+2) = 0 $


$r = -1$:~~  $\left[\matrix{
2 & -1 \cr 6 &-3
 }\right]$$\left[\matrix{
1 \cr 2
 }\right] =  \left[\matrix{0 \cr 0 }\right]$


$r = -2$:~~  $\left[\matrix{
3 & -1 \cr 6 &-2
 }\right]\left[\matrix{
1 \cr 3
 }\right] =  \left[\matrix{0 \cr 0 }\right]$ 



Check: $Av = rv$

$ \left[\matrix{
1 & -1 \cr 6 &-4
 }\right] \left[\matrix{
1 \cr 2
 }\right]  =\left[\matrix{
-1 \cr -2
 }\right] = -1\left[\matrix{
1 \cr 2
 }\right]  $


$ \left[\matrix{
1 & -1 \cr 6 &-4
 }\right] \left[\matrix{
1 \cr 3
 }\right]  =\left[\matrix{
-2 \cr -6
 }\right] = -2\left[\matrix{
1 \cr 3
 }\right]  $



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\centerline{Answer: $\underline{{\bf x} = c_1\left[\matrix{
1 \cr 2 }\right]e^{-t}+ c_2 \left[\matrix{ 1 \cr 3 }\right]e^{-2t}}$}
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[3]~ 1b.)  Describe the behavior of the solution as $t \rightarrow \infty$: $\underline{~~\left[\matrix{x_1 \cr x_2 }\right] \rightarrow \left[\matrix{0 \cr 0 }\right] ~~}$

[7]~ 1c.)  Sketch the phase portrait of this system (i.e. plot a few trajectories of the system).

See class notes (12/8)

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