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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 15 "Phase portraits" }}{PARA 
0 "" 0 "" {TEXT 257 24 "To plot a phase portrait" }}{PARA 0 "" 0 "" 
{TEXT -1 19 "Consider the system" }}{PARA 256 "" 0 "" {OLE 1 4097 1 "[
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:::::::::::::::::::::::::::::::::::::::::::4:" }{TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 53 "The first step is to plot the direction field d
iagram" }}{PARA 0 "" 0 "" {TEXT -1 77 "First load the algebraic and gr
aphics packages linalg and  DEtools as follows" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "
Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warni
ng, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "with(DEtools):" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new defi
nition for adjoint" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Next enter \+
the differential equations. The commands are shown below." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=diff(x(t),t)=y;" }}{PARA 11 "" 
1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "B:=dif
f(y(t),t)=-x-2*y;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 39 "Finally write the command for the plot." }}{PARA 
0 "" 0 "" {TEXT -1 80 "Note that the range -2..5 is the range of t, th
e x and y ranges are explanatory." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "DEplot([A,B],[x(t),y(t)],-2..5,x=-5..5,y=-5..5,lineco
lor=black);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 424 "To convert this into a phase diagram you need to add s
ome trajectories. To do this put in some initial points. First try the
 point (2,3). Note the extra term [x(0)=2,y(0)=3] and its position. Th
e stepsize option controls the stepsize used in the numerical solution
 of the differential equations. It is desirable to use a small enough \+
stepsize to produce a plot with smooth trajectories. This is a matter \+
for experimentation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "DEp
lot([A,B],[x(t),y(t)],-2..5,[[x(0)=2,y(0)=3]],x=-5..5,y=-5..5,linecolo
r=black,stepsize=0.1);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 45 "Now try another one through the point (-2
,-2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "DEplot([A,B],[x(t)
,y(t)],-2..5,[[x(0)=2,y(0)=3],[x(0)=-3,y(0)=-3]],x=-5..5,y=-5..5,linec
olor=black);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 92 "Now try some extra points for yourselves. Try a differ
ent stepsize. Try different ranges of " }{TEXT 258 6 "t, x, " }{TEXT 
-1 4 "and " }{TEXT 259 1 "y" }{TEXT -1 1 "." }}}}{MARK "13" 0 }
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