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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 41 
"Modeling with first order systems of DE's" }}{PARA 258 "" 0 "" {TEXT 
-1 11 "Math 2250-1" }}{PARA 257 "" 0 "" {TEXT -1 25 "Tuesday November \+
25, 2008" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 
18 "Example 1:  Tanks:" }{TEXT -1 374 "  This is example 2 from our te
xt, page 418.  We have a cascade of three tanks, see figure 7.3.2 page
 418.  The tank volumes are V1=20, V2=40, V3=50 (gallons).  The water \+
flow rate is r=10 g/min.  The initial amounts of salt are x1(0)=15, x2
(0)=0,  x3(0)=0.  (You actually had a 2-tank cascade way back in Chapt
er 1, which you solved in steps, without using systems of DEs.)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Show that
 the initial value problem for this system is" }}{PARA 260 "" 0 "" 
{XPPEDIT 18 0 "matrix([[dx1/dt], [dx2/dt], [dx3/dt]]) = matrix([[-.5, \+
0, 0], [.5, -.25, 0], [0, .25, -.2]])*matrix([[x1], [x2], [x3]]);" "6#
/-%'matrixG6#7%7#*&%$dx1G\"\"\"%#dtG!\"\"7#*&%$dx2GF+F,F-7#*&%$dx3GF+F
,F-*&-F%6#7%7%,$-%&FloatG6$\"\"&F-F-\"\"!F>7%-F;6$F=F-,$-F;6$\"#D!\"#F
-F>7%F>-F;6$FEFF,$-F;6$\"\"#F-F-F+-F%6#7%7#%#x1G7#%#x2G7#%#x3GF+" }
{TEXT -1 0 "" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "matrix([[x1(0)], [x2(
0)], [x3(0)]]) = matrix([[15], [0], [0]]);" "6#/-%'matrixG6#7%7#-%#x1G
6#\"\"!7#-%#x2G6#F,7#-%#x3G6#F,-F%6#7%7#\"#:7#F,7#F," }{TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 39 "Solve this initial value problem, using
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "with(linalg):\nDigits:=5
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "A:=matrix([[-.5, 0, 0]
, [.5, -.25, 0], [0, .25, -.2]]):\neigenvectors(A);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%7%$!\"&!\"\"\"\"\"<#-%'vectorG6#7%F'$!&++#!\"%$\"&nm
\"F/7%$!#D!\"#F'<#-F*6#7%\"\"!F'$!&++&F/7%$F5F&F'<#-F*6#7%F:F:F'" }}}
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{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "
You should get" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "matrix([[x1(t)], [x
2(t)], [x3(t)]]) = 5*exp(-.5*t)*matrix([[3], [-6], [5]])+30*exp(-.25*t
)*matrix([[0], [1], [-5]])+125*exp(-.2*t)*matrix([[0], [0], [1]]);" "6
#/-%'matrixG6#7%7#-%#x1G6#%\"tG7#-%#x2GF+7#-%#x3GF+,(*&-%$expG6#,$F,$!
\"&!\"\"\"\"\"-F%6#7%7#\"\"$7#!\"'7#\"\"&F<FE*(\"#IF<-F66#,$F,$!#D!\"#
F<-F%6#7%7#\"\"!7#F<7#F:F<F<*(\"$D\"F<-F66#,$F,$FMF;F<-F%6#7%FQFQFSF<F
<" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 69 "We can plot the three solute amount vs. time to see what \+
is going on:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "x1:=t->15*e
xp(-.5*t):\nx2:=t->-30*exp(-.5*t)+30*exp(-.25*t):\nx3:=t->25*exp(-.5*t
)-150*exp(-.25*t)+125*exp(-.2*t):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "plot(\{x1(t),x2(t),x3(t)\},t=0..30,color=black,\ntitl
e=`salt contents in each tank`);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
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-%&TITLEG6#%;salt~contents~in~each~tankG-%%VIEWG6$;F(Fj_l%(DEFAULTG" 
1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" }}}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 
34 "Example 2:  Glucose-insulin model " }{TEXT -1 108 "  (adapted from
 a discussion on page 339 of the text \"Linear Algebra with Applicatio
ns,\" by Otto Bretscher) " }}{PARA 0 "" 0 "" {TEXT -1 735 "      Let G
(t) be the excess glucose concentration (mg of G per 100 ml of blood, \+
say) in someone's blood, at time t hours.  Excess means we are keeping
 track of the difference between current and equilibrium (\"fasting\")
 concentrations.  Similarly, Let H(t) be the excess insulin concentrat
ion at time t.  When blood levels of glucose rise, say as food is dige
sted, the pancreas reacts by secreting insulin in order to utilize the
 glucose.  Researchers have developed mathematical models for the gluc
ose regulatory system.  Here is a simplified (linearized) version of o
ne such model, with particular representative matrix coefficients.  It
 would be meant to apply between meals, when no additional glucose is \+
being added to the system:" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "matrix(
[[dG/dt], [dH/dt]]) = matrix([[-.1, -.4], [.1, -.1]])*matrix([[G], [H]
]);" "6#/-%'matrixG6#7$7#*&%#dGG\"\"\"%#dtG!\"\"7#*&%#dHGF+F,F-*&-F%6#
7$7$,$-%&FloatG6$F+F-F-,$-F86$\"\"%F-F-7$-F86$F+F-,$-F86$F+F-F-F+-F%6#
7$7#%\"GG7#%\"HGF+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 259 58 "Explain (understand) the signs of the ma
trix coefficients:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 75 "Now let's solve the initial value problem, say right after a bi
g meal, when" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "matrix([[G(0)], [H(0)
]]) = matrix([[100], [0]]);" "6#/-%'matrixG6#7$7#-%\"GG6#\"\"!7#-%\"HG
F+-F%6#7$7#\"$+\"7#F," }{TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A:=matrix(2,2,[-.1,-.4,
.1,-.1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$$!
\"\"F+$!\"%F+7$$\"\"\"F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "eigenvectors(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%^$$!\"\"F&$
\"&++#!\"&\"\"\"<#-%'vectorG6#7$^$$!&++#!\"%$\"\"!F5^$F4$F*F57%^$F%$F2
F)F*<#-F-6#7$^$F1$!\"!F5^$F4$F&F5" }}}{PARA 0 "" 0 "" {TEXT -1 250 "Ca
n you get the same eigenvalues and eigenvectors?  Notice that Maple wr
ites a capital I for the square root of -1, i.  Extract a basis for th
e solution space to his homogeneous system of differential equations f
rom the eigenvector information above:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Solve
 the initial value problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" 
{TEXT -1 71 "Here are some pictures to help understand what the model \+
is predicting:" }}{PARA 0 "" 0 "" {TEXT -1 57 "(1)  Plots of glucose v
s. insulin, at time t hours later:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "G:=t->100*exp(-.1*t)*cos(.2*t):\nH:=t->50*exp(-.1*t)*
sin(.2*t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot(\{G(t),H
(t)\},t=0..30,color=black,title=\n`Excess Glucose and Insulin concentr
ations`);" }}{PARA 13 "" 1 "" {GLPLOT2D 421 226 226 {PLOTDATA 2 "6(-%'
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trationsG-%+AXESLABELSG6$Q\"t6\"Q!Ffhl-%'COLOURG6&%$RGBGF)F)F)-%%VIEWG
6$;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 51 "2) \+
 A phase portrait of the glucose-insulin system:" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 175 "pict1:=fieldplot([-.1*G-.4*H,.1*G-.1*H],G=-40
..100,H=-15..40):\nsoltn:=plot([G(t),H(t),t=0..30],color=black):\ndisp
lay(\{pict1,soltn\},title=`Glucose vs Insulin\nphase portrait`); " }}
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