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{EXCHG {PARA 257 "" 0 "" {TEXT -1 30 "To find the equilibrium point." 
}}{PARA 0 "" 0 "" {TEXT -1 21 "This is clearly (0,0)" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 33 "To classify t
he equilibrium point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wit
h(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for
 norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trac
e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=matrix(2,2,[0,1,-4,
-2]);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 35 "tr(A)= -2, det(A)=4,di
scrim(A)= -12" }}{PARA 0 "" 0 "" {TEXT -1 102 "From the tr-det diagram
 the equilibrium point is a  stable spiral (focus). There will be no d
irectrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "
" {TEXT -1 35 "To plot the direction field diagram" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}{PARA 7 "" 1 "" {TEXT -1 
35 "Warning, new definition for adjoint" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "P:=diff(x(t),t)=y;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Q:=diff(y(t),t)=-4*x-2*y;
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DEplot([P,Q],[x(t),y(t)],-2.
.5,x=-5..5,y=-5..5);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 261 "" 0 "" {TEXT -1 26 "To draw the phase portrait" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 111 "DEplot([P,Q],[x(t),y(t)],-2..5,[[x(0)=4,y(0)=0],[x(0)=-4,y(0)=0
]],x=-5..5,y=-5..5,linecolor=black,stepsize=.1);" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "That the solutio
ns oscillate with decreasing amplitude as they tend to the rest positi
on can be seen by looking at the time series of" }{TEXT 256 2 " x" }
{TEXT -1 9 " against " }{TEXT 257 2 "t." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "DEplot([P,Q],[x(t),y(t)],-1..6,[[x(0)=0,y(0)=5]],x=-
1..1.5,y=-5..5,linecolor=black,stepsize=.1,scene=[t,x]);" }}{PARA 13 "
" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
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