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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 
168 "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 "" }}
{PARA 0 "" 0 "" {TEXT -1 28 "The differential equation is" }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "m*diff(y(t),`$`(t,2))+b*diff(y(t),t)+k*y(t)
 = 0;" "6#/,(*&%\"mG\"\"\"-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F.\"\"#F'F'*&
%\"bGF'-F)6$-F,6#F.F.F'F'*&%\"kGF'-F,6#F.F'F'\"\"!" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 34 "(I've used b instead of the usual " }
{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 34 " for the coefficient
 of friction.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "We convert this to a system by defining" }}{PARA 256 "" 
0 "" {XPPEDIT 18 0 "v(t) = diff(y(t),t);" "6#/-%\"vG6#%\"tG-%%diffG6$-
%\"yG6#F'F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(v(t),t) = -k/m*y(t)-b/m*diff(y(t
),t);" "6#/-%%diffG6$-%\"vG6#%\"tGF*,&*(%\"kG\"\"\"%\"mG!\"\"-%\"yG6#F
*F.F0*(%\"bGF.F/F0-F%6$-F26#F*F*F.F0" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 257 "" 0 "" 
{XPPEDIT 18 0 "diff(v(t),t) = -k/m*y(t)-b/m*v(t);" "6#/-%%diffG6$-%\"v
G6#%\"tGF*,&*(%\"kG\"\"\"%\"mG!\"\"-%\"yG6#F*F.F0*(%\"bGF.F/F0-F(6#F*F
.F0" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "This equation along
 with the equation that defines v(t) make up a system of two equations
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 58 "We first define the two equations th
at make up the system:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eq1 := diff(y(t),t) = v(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/-%%diffG6$-%\"yG6#%\"tGF,-%\"v
GF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eq2 := diff(v(t),t) \+
= -(k/m)*y(t)-(b/m)*v(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/-
%%diffG6$-%\"vG6#%\"tGF,,&*(%\"kG\"\"\"%\"mG!\"\"-%\"yGF+F0F2*(%\"bGF0
F1F2F)F0F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 44 "Now define a variable that holds the system:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sys \+
:= [eq1,eq2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG7$/-%%diffG6$-
%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-,&*(%\"kG\"\"\"%\"mG!\"\"F*F6F8*(%\"bG
F6F7F8F.F6F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 89 "We substitute in some numerical values for the paramete
rs, so we can plot some solutions." }}{PARA 0 "" 0 "" {TEXT -1 127 "(T
he \"subs\" command makes the substitutions given by the first several
 arguments in the expression given as the last argument.)" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sys1
 := subs(b=1/3,m=1,k=1,sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys
1G7$/-%%diffG6$-%\"yG6#%\"tGF--%\"vGF,/-F(6$F.F-,&F*!\"\"*&#\"\"\"\"\"
$F7F.F7F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 161 "Let's first look at the direction field.  This is the sa
me 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=constrained,color=black,title=\"Mass-Spr
ing Direction Field (m=1,k=1,b=1/3)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 
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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "
Let's add a solution curve, also called a " }{TEXT 256 12 "trajectory.
 " }{TEXT -1 187 "We will find the curve with initial condition y(0)=1
, v(0)=0.  The ics argument to DEplot is a list of triples of the form
 [t0,y0,v0].  Here we only put in one initial condition, [0,1,0]." }}
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s1,[y(t),v(t)],t=-5..15,ics,y=-1..1,v=-1..1,scaling=constrained,color=
black,title=\"Mass-Spring Direction Field (m=1,k=1,b=1/3)\",linecolor=
blue,stepsize=.1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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{PARA 0 "" 0 "" {TEXT -1 176 "Let's plot y(t) vs. t for the same solut
ion.  The option \"scene=[var1,var2]\" let's us choose how the solutio
n 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 
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" 0 "" {TEXT -1 82 "Now let's see this solution in the extended phase \+
space.  Here we use the command " }{TEXT 257 8 "DEplot3d" }{TEXT -1 
84 ".  (Don't forget that you can click-and-drag in the 3D plot to see
 different views.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
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