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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 2280-1" }}{PARA 257 "" 0 "" {TEXT -1 23 "Wednesday March 1
, 2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 
18 "Example 1:  Tanks:" }{TEXT -1 274 "  This is example 2 from our te
xt, page 305, and from yesterday's notes.  We have a cascade of three \+
tanks, see figure 5.5.2 The tank volumes are V1=20, V2=40, V3=50 (gall
ons).  The water flow rate is r=10 g/min.  The initial amounts of salt
 are x1(0)=15, x2(0)=0,  x3(0)=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 54 "Show that the initial value problem for t
his system is" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "matrix([[dx1/dt], [d
x2/dt], [dx3/dt]]) = matrix([[-.5, 0, 0], [.5, -.25, 0], [0, .25, -.2]
])*matrix([[x1], [x2], [x3]]);" "6#/-%'matrixG6#7%7#*&%$dx1G\"\"\"%#dt
G!\"\"7#*&%$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$\"\"#F-F-F+-
F%6#7%7#%#x1G7#%#x2G7#%#x3GF+" }{TEXT -1 0 "" }}{PARA 261 "" 0 "" 
{XPPEDIT 18 0 "matrix([[x1(0)], [x2(0)], [x3(0)]]) = matrix([[15], [0]
, [0]]);" "6#/-%'matrixG6#7%7#-%#x1G6#\"\"!7#-%#x2G6#F,7#-%#x3G6#F,-F%
6#7%7#\"#:7#F,7#F," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Solv
e this system, using" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A:=m
atrix([[-.5, 0, 0], [.5, -.25, 0], [0, .25, -.2]]):\neigenvects(A);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6%7%$!#D!\"#\"\"\"<#-%'vectorG6#7%\"\"!
F'$!+++++]!\"*7%$F&!\"\"F'<#-F*6#7%F-F-F'7%$!\"&F3F'<#-F*6#7%F'$!+++++
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{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 "
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
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0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 14 "You should get" }}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'matrixG6#7%7#-%#x1G6#%\"tG7#-%#x2GF+7#-%#x3GF+,(*&-
%$expG6#,$F,$!\"&!\"\"\"\"\"-F%6#7%7#\"\"$7#!\"'7#\"\"&F<FE*(\"#IF<-F6
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%6#7%FQFQFSF<F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "We can plot th
e three solute amount vs. time to see what is going on:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "x1:=t->15*exp(-.5*t):\nx2:=t->-30*
exp(-.5*t)+30*exp(-.25*t):\nx3:=t->25*exp(-.5*t)-150*exp(-.25*t)+125*e
xp(-.2*t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot(\{x1(t),
x2(t),x3(t)\},t=0..30,color=black,\ntitle=`salt contents in each tank`
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\"t6\"Q!Fcim-%'COLOURG6&%$RGBGF)F)F)-%&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" "Curve 2" "Curve 3" }}}}{PAGEBK }
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 34 "Example 2:  Glucose-insul
in model " }{TEXT -1 112 "  (adapted from a discussion on page 340-341
 of the text \"Linear Algebra with Applications,\" 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.  Similar
ly, Let H(t) be the excess insulin concentration at time t.  When bloo
d levels of glucose rise, say as food is digested, the pancreas reacts
 by secreting insulin in order to utilize the glucose.  Researchers ha
ve developed mathematical models for the glucose regulatory system.  H
ere is a simplified (linearized) version of one such model, with parti
cular 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]]) = mat
rix([[-.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 matrix 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 35 "In Bretscher the topic at hand was " }{TEXT 261 26 "discrete dy
namical systems" }{TEXT -1 13 " rather than " }{TEXT 262 11 "continuou
s " }{TEXT -1 30 "ones, and the model given was:" }}{PARA 263 "" 0 "" 
{XPPEDIT 18 0 "matrix([[G(t+1)], [H(t+1)]]) = matrix([[.9, -.4], [.1, \+
.9]])*matrix([[G(t)], [H(t)]]);" "6#/-%'matrixG6#7$7#-%\"GG6#,&%\"tG\"
\"\"F.F.7#-%\"HGF+*&-F%6#7$7$$\"\"*!\"\"$!\"%F97$$F.F9F7F.-F%6#7$7#-F*
6#F-7#-F1FCF." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 138 "Explain
 how I converted this discrete dynamical system into a related first o
rder system of DE's!  (Notice the matrices are NOT the same!)" }}
{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 75 "Now let's solve the initial value problem, say right a
fter a big meal, when" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'ma
trixG6#7$7#-%\"GG6#\"\"!7#-%\"HGF+-F%6#7$7#\"$+\"7#F," }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart
:with(linalg):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 14 "eigenvects(A);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6$7%^$$!\"\"F&$!+++++?!#5\"\"\"<#-%'vectorG6#7$^$$F(!\"*$
!\"!\"\"!^$$F5F5$F&F57%^$F%$\"+++++?F)F*<#-F-6#7$^$F1F7^$F7$F*F5" }}}
{PARA 0 "" 0 "" {TEXT -1 182 "Can you get the same eigenvalues and eig
envectors?  Extract a basis for the solution space to his homogeneous \+
system of differential equations from the eigenvector information abov
e:" }}{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 vs. 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`Exces
s Glucose and Insulin concentrations`);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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