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{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 the
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=matrix(2,2,[0,1,-4,-
5]);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 33 "tr(A)= -5, det(A)=4,dis
crim(A)= 9" }}{PARA 0 "" 0 "" {TEXT -1 95 "From the tr-det diagram the
 equilibrium point is a  stable node. There will be two directrices." 
}}{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 "Warnin
g, 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-5*y;" }}{PARA 
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1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DEplot
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trait" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "First find the two eigen
values" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eigensystem:=eige
nvects(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The directrices are
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" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Ponts on the \+
directrices are (1,-4), (-1,4), (1,-1), (-1,1). Adding some additional
 initial conditions gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "DEplot([P,Q],[x(t),y(t)],-2..5,[[x
(0)=1,y(0)=-4],[x(0)=-1,y(0)=4],[x(0)=1,y(0)=-1],[x(0)=-1,y(0)=1],[x(0
)=4,y(0)=0],[x(0)=-4,y(0)=0],[x(0)=2,y(0)=2],[x(0)=-2,y(0)=-2]],x=-5..
5,y=-5..5,linecolor=black,stepsize=.1);" }}{PARA 13 "" 1 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "That the solutions no lon
ger oscillate as they tend to the rest position 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 0 106 "DEplot(
[P,Q],[x(t),y(t)],-2..5,[[x(0)=0,y(0)=4]],x=-5..5,y=-5..5,linecolor=bl
ack,stepsize=.1,scene=[t,x]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "5 0 0" 10 }
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