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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
edefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Parameterized curve
s." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "x:=t->sqrt
(1+sin(t^(1.1)));\ny:=t->sqrt(t)+1/(sin(t)+1.5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"xGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\"
\"\"F0-%$sinG6#*$)9$$\"#6!\"\"F0F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"yGR6#%\"tG6\"6$%)operatorG%&arrowGF(,&-%%sqrtG6#9$\"\"\"*&F1
F1,&-%$sinGF/F1$\"#:!\"\"F1F8F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "plot(\{x(t),y(t)\},t=0..20);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "plot([x(t),y(t),t=0..20]);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 65 "animate([x(t*a),y(t*a),a=0..1],t=0..20,frames
=100,numpoints=300);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "This particle moves in a circle" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot([cos(2*Pi*t),sin(2*Pi*t),t=0..
1/2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "animate([cos(2*Pi
*t*a),sin(2*Pi*t*a),a=0..1],t=0..1/2,frames=100,numpoints=300);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Consider a pa
rticle moving around described by the following equations:" }{MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x:=t->t^2;\ny:=t->t^3;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGR6#%\"tG6\"6$%)operatorG%&arro
wGF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGR6
#%\"tG6\"6$%)operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(\{x(t),y(t)\},t=-1..1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([x(t),y(t),t=-1..1],sca
ling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Can we fig
ure out how fast it is going? How fast is it moving in the x direction
? How fast is it moving in the y direction? Is it ever stopped?" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 227 "The rate of change of the particl
e in the x direction is just the derivative of x(t). Similarly its spe
ed in the y direction is just the derivative of y(t). To find out if i
t is ever stopped we just need to solve the equations " }}{PARA 0 "" 
0 "" {XPPEDIT 18 0 "diff(x,t) = 0,diff(y,t) = 0;" "6$/-%%diffG6$%\"xG%
\"tG\"\"!/-F%6$%\"yGF(F)" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "Eq:=\{diff(x(t),t)=0,diff(y(t),t)=0\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#EqG<$/,$%\"tG\"\"#\"\"!/,$*$)F(F)\"\"\"\"\"$F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(Eq,t);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<#/%\"tG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "animate([x(t*a+(-1)*(1-a)),y(t*a+(-1)*(1-a)),a=0..1],
t=-1..1,frames=100,numpoints=300);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "x:=t->cos(2*Pi*t);\ny:=t->si
n(2*Pi*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGR6#%\"tG6\"6$%)ope
ratorG%&arrowGF(-%$cosG6#,$*&%#PiG\"\"\"9$F2\"\"#F(F(F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"yGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$sinG6
#,$*&%#PiG\"\"\"9$F2\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 64 "animate([x(t),a*y(t),a=0..1],t=0..2*Pi,frames=300,numpoints=10
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 
"fieldplot([-y*2*Pi,x*2*Pi],x=-1..1,y=-1..1);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 34 "Systems of differential equations." }{MPLTEXT 
1 0 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "FP:=fieldplot([y+.1*x,
-.4*x],x=-1..1,y=-1..1,scaling=constrained):display(FP);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 117 "We could write these \"moving directions
\" by saying that particles x coordinate has to be changing at a rate \+
given by\n" }{XPPEDIT 18 0 "diff(x,t) = y+.1*x;" "6#/-%%diffG6$%\"xG%
\"tG,&%\"yG\"\"\"*&$F+!\"\"F+F'F+F+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 69 "and the particles y coordinate has to be changing at a ra
te given by\n" }{XPPEDIT 18 0 "diff(y,t) = -.4*x;" "6#/-%%diffG6$%\"yG
%\"tG,$*&$\"\"%!\"\"\"\"\"%\"xGF.F-" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 77 "Note that both x and y vary with time, so x and y are bot
h functions of time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 76 "We can make maple sovle these differential equations ex
plicitly with dsolve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq
:=\{diff(x(t),t)=y(t)+.1*x(t),diff(y(t),t)=-.4*x(t)\};\n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#eqG<$/-%%diffG6$-%\"xG6#%\"tGF-,&-%\"yGF,\"\"
\"*&$F1!\"\"F1F*F1F1/-F(6$F/F-,$F*$!\"%F4" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 128 "Note that we have to explicitly tell maple that x and y \+
depend on t (which is time) by writing x(t) and y(t) instead of x and \+
y." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Ex
Sols:=dsolve(eq,\{x(t),y(t)\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'
ExSolsG<$/-%\"xG6#%\"tG,$*&-%$expG6#,$F*#\"\"\"\"#?F2,**&%$_C1GF2-%$co
sG6#,$*&-%%sqrtG6#\"$f\"F2F*F2F1F2!\"\"*(F6F2-%$sinGF9F2F<F2F2*&%$_C2G
F2FBF2F@*(FEF2F7F2F<F2F@F2#F2\"\")/-%\"yGF)*&F-F2,&F5F2FDF2F2" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 351 "This gives us explicit solutions.
 There are actually infinitely many solutions, because there are infin
itely many starting points for the particle, and each gives rise to a \+
distinct solution. Specifying values for the constants _C1 and _C2 giv
es a specific solution. We could then plot this for different values o
f _C1 and _C2 to see solution curves. " }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 243 "Most differential equations are very har
d ot solve explicitly, or the explicit solution is so complicated that
 it isn't very helpful. Maple has another command for plotting solutio
ns directly, without coming up with formulas for the solution. " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 50 "DEplot(eq,[x(t),y(t)],t=0..10,[[x(0)=.5,y(0)
=0]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 317 "Maple creates this plo
t by following the arrows in a field plot, taking consecutive small, s
traight steps, each representing a small amount of time that has passe
d while the particle moves.. You can see the steps in the last plot. T
o eliminate this change the stepsize (the amount of time each step las
ts) as follows:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "DEplot(eq,[x(t),y(t)],t=0..10,[[x(0)=.5,y(0)=0]],step
size=.1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The fieldplot comman
d produces better arrows. We can turn off the DEplot arrows as follows
:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "DEplot(eq,[
x(t),y(t)],t=0..10,[[x(0)=.5,y(0)=0]],stepsize=.1,arrows=none);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Now we can combine this with the e
arlier fieldplot." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 96 "PL:=DEplot(eq,[x(t),y(t)],t=0..10,[[x(0)=.5,y(0)=0]],stepsize=.1
,arrows=none):\ndisplay(\{FP,PL\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 219 "From this we can see that the solution spirals away from the o
rigin. We can see more looking at the explicit solution above, namely \+
when we write out formulas for x and y they are each a periodic functi
on multiplied by " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "e^(t/20);" "6#)%\"
eG*&%\"tG\"\"\"\"#?!\"\"" }{TEXT -1 123 " and thus they grow exponenti
ally with time (and always decay exponentially toward the origin if we
 look backward in time)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 49 "Equilibria points (\"Nonlinear Systems\" page 176).
" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 82 "Stable/unstable node, stable/unstable foc
us (spiral point), center, saddle point. " }{MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "S
table/unstable curve of a saddle point." }{MPLTEXT 1 0 0 "" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "35 0 0" 3 }{VIEWOPTS 1 1 0 2 1 
1805 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }