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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 22 "The Mass-Spring System" }
}{PARA 19 "" 0 "" {TEXT -1 8 "Math 308" }}{PARA 0 "" 0 "" {TEXT -1 
169 "This Maple session uses the mass-spring system to demonstrate the
 phase plane, direction fields, solution curves (``trajectories''), th
e extended phase space, and more. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We first \+
define the two equations that make up the system:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eq1 := diff
(y(t),t) = v(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/-%%diffG6$
-%\"yG6#%\"tGF,-%\"vGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "
eq2 := diff(v(t),t) = -(b/m)*v(t)-(k/m)*y(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq2G/-%%diffG6$-%\"vG6#%\"tGF,,&*(%\"bG\"\"\"%\"mG!
\"\"F)F0F2*(%\"kGF0F1F2-%\"yGF+F0F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Now define a variable that hold
s the system:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "sys := [eq1,eq2];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$sysG7$/-%%diffG6$-%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-,&*(%\"bG\"
\"\"%\"mG!\"\"F.F6F8*(%\"kGF6F7F8F*F6F8" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "We substitute in some num
erical values for the parameters, so we can plot some solutions." }}
{PARA 0 "" 0 "" {TEXT -1 127 "(The \"subs\" command makes the substitu
tions given by the first several arguments in the expression given as \+
the last argument.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "sys1 := subs(b=1/3,m=1,k=1,sys);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys1G7$/-%%diffG6$-%\"yG6#%\"tGF--%
\"vGF,/-F(6$F.F-,&F.#!\"\"\"\"$F*F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "Let's first look at the direct
ion field.  This is the same as the vector field, but all the vectors \+
are plotted with the same length.  We use the DEplot command." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 132 "DEplot(sys1,[y(t),v(t)],t=0..1,y=-1..1,v=-1..1,scaling=constrai
ned,color=black,title=`Mass-Spring Direction Field (m=1,k=1,b=1/3)`);
" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6]
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45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
176 "Let's plot y(t) vs. t for the same solution.  The option \"scene=
[var1,var2]\" let's us choose how the solution is plotted.  If we say \+
\"scene=[t,y]\", we get a plot of y(t) vs. t." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 147 "DEplot(sys1,[y(t),v(t)],t=-5..15,ics,y=-2.5..1.
5,v=-1..1,color=black,title=`Mass-Spring   (m=1,k=1,b=1/3)`,linecolor=
blue,stepsize=.1,scene=[t,y]);" }}{PARA 13 "" 1 "" {GLPLOT2D 427 236 
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