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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 9 "John Cole" }}{PARA 256 "
" 0 "" {TEXT -1 14 "Course Work: 2" }}{PARA 256 "" 0 "" {TEXT -1 9 "Pr
oblem 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Investigate the nature
 and stability of the equilibrium point of the following systems:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "amy := diff(x(t),t) = 3*x(t)
+alpha*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$amyG/-%%diffG6$-%\"
xG6#%\"tGF,,&F)\"\"$-%\"yGF+\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "anita := diff(y(t),t) = -x(t)+5*y(t);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&anitaG/-%%diffG6$-%\"yG6#%\"tGF,,&-%\"xGF+!\"\"
*&\"\"&\"\"\"F)F3F3" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 13 "A). alph
a > 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&alphaG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "lambda1 := e
igenvectors([[3,2],[-1,5]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lam
bda1G6$7%^$\"\"%\"\"\"F)<#-%'vectorG6#7$^$F)!\"\"F)7%^$F(F0F)<#-F,6#7$
^$F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The first eigenvalue \+
and corresponding vector is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "lambda1[1][1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lambda1[1][
3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$\"\"%\"\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#<#-%'vectorG6#7$^$\"\"\"!\"\"F)" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 50 "The second eigenvalue and corresponding vector is:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lambda1[2][1];" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lambda1[2][3];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#^$\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'ve
ctorG6#7$^$\"\"\"F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "We can \+
see that when alpha is greater than 1 (in this case 2) the resulting e
igenvalues for the system are positive real and positiv/negative imagi
nary.  The systems equilibrium point is a spiral and it is unstable.  \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 13 "B). alpha = 1
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "lambda2 := eigenvectors([[3,
1],[-1,5]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda2G7%\"\"%\"\"
#<#-%'vectorG6#7$\"\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The \+
first eigenvalue is equal to the second eigenvalue, which is 4. " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Analysis:  The equilibrium point i
s a proper or improper node and the system is unstable." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 257 14 "C).  alpha < 1" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "lambda3 := eigenvectors([[3,0],[-1,5]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(lambda3G6$7%\"\"$\"\"\"<#-%'vectorG6#7$\"\"#F(7%\"\"
&F(<#-F+6#7$\"\"!F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The first \+
eignevalue is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lambda3[1
][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 25 "The second eigenvalue is:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 14 "lambda3[2][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 11 "Analysis:  " }{TEXT -1 
178 "We can see that both vectors are entirely real and positive, howe
ver not equal.  Thus when alpha is less than 1, the system's critical \+
point is a node and the system is unstable." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 259 16 "D).  alpha = -15" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "lambda4 := eigenvectors([[3,-15],[-1,5]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(lambda4G6$7%\"\")\"\"\"<#-%'vectorG6#7$!\"$F(7%\"\"!
F(<#-F+6#7$\"\"&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The first e
ignevalue is:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "lambda4[1][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The second eigenvalue is:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lambda4[2][1];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Anal
ysis:  ????????????" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT -1 15 "E). alpha < -15" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "lambda5 := eigenvectors([[3,-20],[-1,5]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%(lambda5G6$7%,&\"\"%\"\"\"*$-%%sqrtG6#\"#@
F)F)F)<#-%'vectorG6#7$,&F)F)F*!\"\"F)7%,&F(F)F*F5F)<#-F16#7$,&F)F)F*F)
F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The first eigenvalue is:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lambda5[1][1];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&\"\"%\"\"\"*$-%%sqrtG6#\"#@F%F%" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 25 "The second eigenvalue is:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "lambda5[2][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
\"\"%\"\"\"*$-%%sqrtG6#\"#@F%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 173 "Analysis:  We can see that both lambdas are real numbers.  Lam
bda1 is positive and lambda2 is negative.  Therefore the critical poin
t is a saddle point and it is unstable.  " }{MPLTEXT 1 0 0 "" }}}}
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