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:6:" }}}{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 cl
assify the equilibrium point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new defin
ition for adjoint" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A:=mat
rix(2,2,[0,1,-4,0]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 259 "" 0 "" {TEXT -1 33 "tr(A)=0, det(A)=4,discrim(A)= -16" }}
{PARA 0 "" 0 "" {TEXT -1 74 "From the tr-det diagram the equilibrium p
oint is a neutrally stable centre" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 260 "" 0 "" {TEXT -1 35 "To plot the direction field diag
ram" }}}{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 21 "
Q:=diff(y(t),t)=-4*x;" }}{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" }}{PARA 0 "" 0 "" 
{TEXT -1 55 "There are no directrices. Putting in initial conditions" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "DEplot([P,Q],[x(t),y(t)]
,-2..5,[[x(0)=0,y(0)=2],[x(0)=0,y(0)=4]],x=-5..5,y=-5..5,linecolor=bla
ck,stepsize=.1);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 113 "That the solutions are periodic oscillating with \+
constant amplitude can be seen by looking at some time series of" }
{TEXT 256 2 " x" }{TEXT -1 9 " against " }{TEXT 257 2 "t." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "DEplot([P,Q],[x(t),y(t)],-2..5,[[x
(0)=0,y(0)=2],[x(0)=0,y(0)=4]],x=-5..5,y=-5..5,linecolor=black,stepsiz
e=.1,scene=[t,x]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
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