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{SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT -1 47 "Chapter 2: Systems of two
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0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 84 "Since it's hard to see \+
what's going on near an equilibrium point (because the arrows" }}
{PARA 0 "" 0 "" {TEXT -1 85 "get so small), sometimes it's better to s
crap the speed information, and make all the" }}{PARA 0 "" 0 "" {TEXT 
-1 129 "arrows the same length. TO do this, we just divide the vector \+
field by its own length:\nThat's what we get with a direction field:" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "f1:=x-y:\nf2:=x+y:\nlen:=s
qrt(f1^2+f2^2):\nfieldplot( [f1/len,f2/len], x=-1..1,y=-1..1);" }}
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etF\\^u7$$\"3!pi(>')3HU<FetF_^u7$$\"3.__34C&))G#FetFb^u7%7$Ff]vF[`t7$F
i]vF^`t7$F\\^vFa`t7%7$F`jvF[bs7$FcjvF^bs7$$\"3(>_$[v9cnBF-Fabs7%7$Ffew
F\\cr7$FiewF_cr7$F\\fwFbcr7%7$Fd`xFfcq7$Fg`xFicq7$Fj`xF\\dq7%7$F^jxF^d
p7$FajxFadp7$FdjxFddp7%7$F`dyFjco7$FcdyF]do7$FfdyF`do7%7$F^]zFjbn7$Fa]
zF]cn7$Fd]zF`cn7%7$FbdzF\\am7$FedzF_am7$FhdzFbam7%7$F`[[lFd^l7$F`[[lFg
^l7$Ff[[l$\"3#[^UnVT)H5F1-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 
1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 29 "Euler's method f
or 2D systems" }}{PARA 0 "" 0 "" {TEXT -1 173 "Here is how use Maple t
o perform Euler's method for a 2D system.\nIt's a straightfoward exten
sion of the 1D version.\nI'm applying it to the Van der Pol initial va
lue problem:" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"
xG6#%\"tGF*%\"yG" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6
$-%\"yG6#%\"tGF*,&%\"xG!\"\"*&,&\"\"\"F0*$F,\"\"#F-F0F(F0F0" }}}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#\"\"!\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 542 "dt:=0.25:\nn:=ceil(1.0/dt):  \nt:=0.0:  \nx:=
1.0:  \ny:=1.0:  \ntxdata[0] := [t,x]:  \ntydata[0] := [t,y]:  \nxydat
a[0] := [x,y]:  \nfor i from 1 to n do     \n   mk := y:     \n   nk :
= -x + (1-x^2)*y:     \n   x := x + dt*mk:     \n   y := y + dt*nk:   \+
  \n   t := t + dt:     \n   txdata[i] := [t,x]:   \n   tydata[i] := [
t,y]:  \n   xydata[i] := [x,y]:  \nod:  \nt:='t': x:='x': y:='y':     \+
                 # (clean-up)\ntxdatalist := [ seq( txdata[i], i=0..n \+
) ]:  \ntydatalist := [ seq( tydata[i], i=0..n ) ]:  \nxydatalist := [
 seq( xydata[i], i=0..n ) ]:" }}}{PARA 0 "" 0 "" {TEXT -1 66 "If we do
n't suppress the output on the last three lines, we'll see" }}{EXCHG 
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+txdatalistG7'7$\"\"!$\"#5!\"\"7$$\"
#D!\"#$\"%]7!\"$7$$\"#]F.$\"']P9!\"&7$$\"#vF.$\"+8y]?:!\"*7$$\"$+\"F.$
\"+j=a\"\\\"F=" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+tydatalistG7'7$\"
\"!$\"#5!\"\"7$$\"#D!\"#$\"$](!\"$7$$\"#]F.$\"++DJ?L!#57$$\"#vF.$!++!Q
'e6F77$$\"$+\"F.$!+)[!*)zXF7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+xyd
atalistG7'7$$\"#5!\"\"F'7$$\"%]7!\"$$\"$](F-7$$\"']P9!\"&$\"++DJ?L!#57
$$\"+8y]?:!\"*$!++!Q'e6F67$$\"+j=a\"\\\"F:$!+)[!*)zXF6" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "You can make time-pl
ots of the dependent variables like this:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "plot( \{txdatalist,tydatalist\}, \n`t`=0..1.2, `x and
 y`=-0.5..1.6);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7'7$\"\"!$
\"\"\"F(7$$\"1+++++++D!#;$\"1++++++]7!#:7$$\"1+++++++]F.$\"1+++++]P9F1
7$$\"1+++++++vF.$\"1+++8y]?:F17$F)$\"1+++j=a\"\\\"F1-%'COLOURG6&%$RGBG
$\"#5!\"\"F(F(-F$6$7'F'7$F,F87$F3$\"1++++DJ?LF.7$F8$!1++++!Q'e6F.7$F)$
!1+++)[!*)zXF.-F@6&FBF(FCF(-%+AXESLABELSG6$%\"tG%(x~and~yG-%%VIEWG6$;F
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{TEXT -1 26 "and phase plots like this:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "plot( xydatalist );" }}{PARA 13 "" 1 "" {INLPLOT "6#-
%'CURVESG6$7'7$$\"\"\"\"\"!F(7$$\"1++++++]7!#:$\"1+++++++v!#;7$$\"1+++
++]P9F.$\"1++++DJ?LF17$$\"1+++8y]?:F.$!1++++!Q'e6F17$$\"1+++j=a\"\\\"F
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{PARA 0 "" 0 "" {TEXT -1 114 "You can combine the Euler x,y data plot \+
with a direction field using \"display\" from the \"plots\" package, a
s before" }}{PARA 0 "" 0 "" {TEXT -1 16 "(see Ch1 guide)." }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT 256 29 "Plotting parametrized curves." }}
{PARA 0 "" 0 "" {TEXT -1 53 "Here is the syntax for plotting a paramet
rized curve:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "x:=exp(-3*t)
;\ny:=exp(-t);\nplot( [x,y,t=0..2] );" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"xG-%$expG6#,$%\"tG!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"y
G-%$expG6#,$%\"tG!\"\"" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7W7
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "This form of the plot command is \+
also useful for plotting a function y(t) when you actually have t(y):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "restart:\nt:=9*y+8*cos(
y);\nplot([t,y,y=0..3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG,&%
\"yG\"\"*-%$cosG6#F&\"\")" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$
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pi#F.7$$\"1UE\"pa$R)p\"F@$\"1+++bJ*[o#F.7$$\"1lV$y[gjt\"F@$\"1++Dr\"[8
v#F.7$$\"1Zc,&4$3t<F@$\"1+++Ijy5GF.7$$\"1(*G>c==:=F@$\"1+]P/)fT(GF.7$$
\"1S*=zSw$e=F@$\"1+]i0j\"[$HF.7$$\"1W'>F+1!3>F@$\"\"$F*-%'COLOURG6&%$R
GBG$\"#5!\"\"F*F*-%+AXESLABELSG6$%!GFc[l-%%VIEWG6$%(DEFAULTGFg[l" 2 
305 305 305 2 0 1 0 2 9 0 4 2 1 0.000000 45.000000 45.000000 10030 
10061 10056 10074 0 0 0 20530 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 2 0 0 0 
0 0 0 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Note that y is on \+
the vertical axis - where we want it!" }}}{EXCHG {PARA 0 "> " 0 "" 
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1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }