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{SECT 0 {PARA 205 "" 0 "" {TEXT 219 36 "Modeling with Differential Equ
ations" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 220 8 "restart;" }}}
{SECT 0 {PARA 207 "" 0 "" {TEXT 221 34 "Two Different Models for Free \+
Fall" }}{SECT 1 {PARA 207 "" 0 "" {TEXT 221 7 "Model I" }}{PARA 206 "
" 0 "" {TEXT 222 85 "Newton's second law states that force is equal to
 mass times acceleration, i.e.\n     " }{XPPEDIT 18 0 "F = m;" "6#/%\"
FG%\"mG" }{TEXT 223 2 "  " }{TEXT 222 1 " " }{XPPEDIT 18 0 "dv/dt" "6#
*&%#dvG\"\"\"%#dtG!\"\"" }{TEXT 223 2 "  " }{TEXT 222 3 "  ." }}{PARA 
206 "" 0 "" {TEXT 222 91 "Near the earth's surface, the force due to g
ravity is the weight of the object, i.e. \n     " }{XPPEDIT 18 0 "F = \+
m*g" "6#/%\"FG*&%\"mG\"\"\"%\"gGF'" }{TEXT 223 2 "  " }{TEXT 222 118 "
.\nIf we assume we are in a vacuum and no other forces are acting, the
n this leads to the simple differential equation " }}{PARA 206 "" 0 "
" {TEXT 222 5 "     " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 223 2 "  " }
{TEXT 222 1 " " }{XPPEDIT 18 0 "dv/dt = m*g" "6#/*&%#dvG\"\"\"%#dtG!\"
\"*&%\"mGF&%\"gGF&" }{TEXT 223 2 "  " }{TEXT 222 10 "    or    " }
{XPPEDIT 18 0 "dv/dt = g" "6#/*&%#dvG\"\"\"%#dtG!\"\"%\"gG" }{TEXT 
223 2 "  " }{TEXT 222 1 "." }}{PARA 206 "" 0 "" {TEXT 222 30 "The solu
tion of this model is " }{XPPEDIT 18 0 "v(t) = g*t + C" "6#/-%\"vG6#%
\"tG,&*&%\"gG\"\"\"F'F+F+%\"CGF+" }{TEXT 223 2 "  " }{TEXT 222 54 ". A
nd if the initial velocity is equal to zero, i.e., " }{XPPEDIT 18 0 "v
(0)=0" "6#/-%\"vG6#\"\"!F'" }{TEXT 223 2 "  " }{TEXT 222 7 ", then " }
{XPPEDIT 18 0 "C=0" "6#/%\"CG\"\"!" }{TEXT 223 2 "  " }{TEXT 222 5 " a
nd " }}{PARA 206 "" 0 "" {TEXT 222 5 "     " }{XPPEDIT 18 0 "v(t) = g*
t" "6#/-%\"vG6#%\"tG*&%\"gG\"\"\"F'F*" }{TEXT 223 2 "  " }{TEXT 222 1 
"." }}{PARA 206 "" 0 "" {TEXT 222 33 "We can also use Maple's built-in
 " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT 222 31 " command to obt
ain this answer." }}{PARA 206 "" 0 "" {TEXT 222 23 "This is how it's d
one: " }}{PARA 206 "" 0 "" {HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT 
222 116 " accepts as parameters the differential equation together wit
h an optional initial condition (both enclosed in \{\}). " }}{PARA 
206 "" 0 "" {TEXT 222 83 "Note that the derivatives in the differentia
l equations have to be specified using " }{HYPERLNK 17 "diff" 2 "diff
" "" }{TEXT 222 3 " . " }}{PARA 206 "" 0 "" {TEXT 222 24 "The second a
rgument for " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT 222 59 " is \+
the function to be solved for (including its variable)." }}{EXCHG 
{PARA 206 "> " 0 "" {MPLTEXT 1 220 46 "dsolve(\{m*diff(v(t), t) = m*g,
 v(0)=0\}, v(t));" }{TEXT -1 0 "" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 
222 190 "If we want to further process this solution we must first tel
l Maple to actually assign the right-hand side of the output to the le
ft-hand side (note that the output says \"=\" and not \":=\"). " }}
{PARA 206 "" 0 "" {TEXT 222 26 "This can be done with the " }
{HYPERLNK 17 "assign" 2 "assign" "" }{TEXT 222 53 " command. (Note tha
t this command produce no output)." }}{PARA 206 "> " 0 "" {MPLTEXT 1 
220 17 "assign(v=rhs(%));" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 222 4 "No
w " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT 223 2 "  " }{TEXT 222 8 " is t
he " }{TEXT 225 10 "expression" }{TEXT 222 1 " " }{XPPEDIT 18 0 "g*t" 
"6#*&%\"gG\"\"\"%\"tGF%" }{TEXT 223 2 "  " }{TEXT 222 27 " and we can \+
turn it into a " }{TEXT 225 8 "function" }{TEXT 222 7 " using " }
{HYPERLNK 17 "unapply" 2 "unapply" "" }{TEXT 222 2 " ." }}{PARA 206 ">
 " 0 "" {MPLTEXT 1 220 22 "v;\nv := unapply(v, t);" }{TEXT -1 0 "" }}}
{EXCHG {PARA 206 "" 0 "" {TEXT 222 6 "Since " }{XPPEDIT 2 0 "v" "6#%\"
vG" }{TEXT 223 1 " " }{TEXT 222 74 " is now a function, we can easily \+
evaluate the solution at any given time " }{XPPEDIT 2 0 "t" "6#%\"tG" 
}{TEXT 223 1 " " }{TEXT 222 47 " or create a plot over an entire time \+
interval." }}{PARA 206 "> " 0 "" {MPLTEXT 1 220 75 "g := 9.81;\n'v(10)
' = v(10);\nplot(v(t), t=0..10, title=\"Velocity vs. Time\");" }{TEXT 
-1 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 
207 "" 0 "" {TEXT 221 8 "Model II" }}{PARA 206 "" 0 "" {TEXT 222 98 "L
et us now make the model slightly more realistic and include some effe
cts due to air resistance. " }}{PARA 206 "" 0 "" {TEXT 222 125 "We wil
l assume that air resistance is proportional to the velocity and that \+
it acts in the opposite direction of the motion. " }}{PARA 206 "" 0 "
" {TEXT 222 55 "This results in a new differential equation of the for
m" }}{PARA 206 "" 0 "" {TEXT 222 7 "       " }{XPPEDIT 18 0 "m" "6#%\"
mG" }{TEXT 223 3 "   " }{XPPEDIT 18 0 "dv/dt = m*g-k*v;" "6#/*&%#dvG\"
\"\"%#dtG!\"\",&*&%\"mGF&%\"gGF&F&*&%\"kGF&%\"vGF&F(" }{TEXT 223 4 "  \+
, " }}{PARA 206 "" 0 "" {TEXT 222 6 "where " }{XPPEDIT 18 0 "k" "6#%\"
kG" }{TEXT 223 36 "   is a constant of proportionality." }}{PARA 206 "
" 0 "" {TEXT 222 79 "Now simple integration is no longer suffcient to \+
solve this problem (Try it!). " }}{PARA 206 "" 0 "" {TEXT 222 23 "Howe
ver, using Maple's " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT 222 
73 " (or some of the techniques we will learn shortly), we can still d
o this." }}{EXCHG {PARA 206 "" 0 "" {TEXT 222 86 "First we need to res
tart Maple (or at least break some of the assignments made above)." }}
{PARA 206 "> " 0 "" {MPLTEXT 1 220 8 "restart;" }}}{EXCHG {PARA 206 "
" 0 "" {TEXT 222 12 "The call to " }{HYPERLNK 17 "dsolve" 2 "dsolve" "
" }{TEXT 222 29 " is almost the same as before" }}{PARA 206 "> " 0 "" 
{MPLTEXT 1 220 55 "dsolve(\{m*diff(v(t), t) = m*g - k*v(t), v(0)=0\}, \+
v(t));" }{TEXT -1 0 "" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 222 41 "Again
, to enable further processing we do" }}{PARA 206 "> " 0 "" {MPLTEXT 
1 220 37 "assign(v=rhs(%));\nv := unapply(v, t);" }{TEXT -1 0 "" }}}
{EXCHG {PARA 206 "" 0 "" {TEXT 222 125 "And here is a plot for the cas
e where the mass is 10 kg, and the air resistance is equal to twice th
e instantaneous velocity." }}{PARA 206 "> " 0 "" {MPLTEXT 1 220 119 "m
 := 10: k := 2: g := 9.81:\n'v(10)' = evalf(v(10));\nplot(v(t), t=0..1
0, title=\"Velocity vs. Time with Air Resistance\");" }{TEXT -1 0 "" }
}}{PARA 206 "" 0 "" {TEXT 222 45 "Note that with this model the veloci
ty after " }{XPPEDIT 18 0 "t=10" "6#/%\"tG\"#5" }{TEXT 223 19 "   seco
nds is 42.4 " }{XPPEDIT 18 0 "m/s" "6#*&%\"mG\"\"\"%\"sG!\"\"" }{TEXT 
223 46 "  , whereas the previous model predicted 98.1 " }{XPPEDIT 18 
0 "m/s" "6#*&%\"mG\"\"\"%\"sG!\"\"" }{TEXT 223 41 "   (independent of \+
the mass of the body)." }}{EXCHG {PARA 208 "" 0 "" {TEXT -1 0 "" }}}}}
{SECT 1 {PARA 209 "" 0 "" {TEXT 226 48 "Foxes and Rabbits (The Lotka-V
olterra Equations)" }}{PARA 208 "" 0 "" {TEXT -1 0 "" }}{PARA 206 "" 
0 "" {TEXT 222 127 "A standard model for describing an ecosystem in wh
ich two competing species exist (one a predator, the other its prey) a
re the " }{TEXT 225 24 "Lotka-Volterra equations" }}{PARA 206 "" 0 "" 
{TEXT 222 4 "    " }{XPPEDIT 18 0 "dx/dt = -a*x+b*x*y" "6#/*&%#dxG\"\"
\"%#dtG!\"\",&*&%\"aGF&%\"xGF&F(*(%\"bGF&F,F&%\"yGF&F&" }{TEXT 223 7 "
  \n    " }{XPPEDIT 18 0 "dy/dt = d*y - c*x*y" "6#/*&%#dyG\"\"\"%#dtG!
\"\",&*&%\"dGF&%\"yGF&F&*(%\"cGF&%\"xGF&F,F&F(" }{TEXT 223 4 "  , " }}
{PARA 206 "" 0 "" {TEXT 222 6 "where " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG
6#%\"tG" }{TEXT 223 7 "   and " }{XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG
" }{TEXT 223 77 "  , respectively, denote the population of the predat
or and the prey at time " }{XPPEDIT 2 0 "t" "6#%\"tG" }{TEXT 223 3 " ,
 " }}{PARA 206 "" 0 "" {TEXT 223 4 "and " }{XPPEDIT 18 0 "a" "6#%\"aG
" }{TEXT 223 4 "  , " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT 223 4 "  , \+
" }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT 223 8 "  , and " }{XPPEDIT 18 0 
"d" "6#%\"dG" }{TEXT 223 84 "   are positive constants representing ef
fects such as birth rate, death rate, etc.." }}{PARA 206 "" 0 "" 
{TEXT 222 70 "The two differential equations are based on the followin
g assumptions:" }}{PARA 210 "" 0 "" {TEXT 227 115 "If there is no prey
, then the population of the predators should be decreasing due to ina
dequate food supply, thus " }{XPPEDIT 18 0 "dx/dt=-a*x" "6#/*&%#dxG\"
\"\"%#dtG!\"\",$*&%\"aGF&%\"xGF&F(" }{TEXT 228 3 "  ." }}{PARA 210 "" 
0 "" {TEXT 227 210 "If prey is present, then the population of the pre
dators will increase, but the rate of increase will be proportional to
 the number of interactions between the two species (which is proporti
onal to the product " }{XPPEDIT 18 0 "x*y" "6#*&%\"xG\"\"\"%\"yGF%" }
{TEXT 228 50 "  ), thus (together with the previous assumption) " }
{XPPEDIT 18 0 "dx/dt = -a*x + b*x*y" "6#/*&%#dxG\"\"\"%#dtG!\"\",&*&%
\"aGF&%\"xGF&F(*(%\"bGF&F,F&%\"yGF&F&" }{TEXT 228 3 "  ." }}{PARA 210 
"" 0 "" {TEXT 227 169 "If there are no predators, then the population \+
of the prey should grow (assuming unlimited food supply and space) at \+
a rate proportional to the present population, i.e. " }{XPPEDIT 18 0 "
dy/dt = d*y" "6#/*&%#dyG\"\"\"%#dtG!\"\"*&%\"dGF&%\"yGF&" }{TEXT 228 
3 "  ." }}{PARA 210 "" 0 "" {TEXT 227 81 "If there are predators, then
 the prey population should decrease proportional to " }{XPPEDIT 18 0 
"x*y" "6#*&%\"xG\"\"\"%\"yGF%" }{TEXT 228 50 "  , i.e., (together with
 the previous assumption) " }{XPPEDIT 18 0 "dy/dt=d*y-c*x*y" "6#/*&%#d
yG\"\"\"%#dtG!\"\",&*&%\"dGF&%\"yGF&F&*(%\"cGF&%\"xGF&F,F&F(" }{TEXT 
228 4 "  .\n" }}{PARA 206 "" 0 "" {TEXT 222 58 "We now solve the follo
wing problem using Maple's built-in " }{HYPERLNK 17 "dsolve" 2 "dsolve
" "" }{TEXT 222 71 " command thay can also be applied to systems of di
fferential equations:" }}{PARA 206 "" 0 "" {TEXT 222 15 "Assume at tim
e " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT 223 95 "   there are 4
 foxes and 4 rabbits, and that the constants in the Lotka-Volterra equ
ations are " }{XPPEDIT 18 0 "a=0.16" "6#/%\"aG-%&FloatG6$\"#;!\"#" }
{TEXT 223 4 "  , " }{XPPEDIT 18 0 "b=0.08" "6#/%\"bG-%&FloatG6$\"\")!
\"#" }{TEXT 223 4 "  , " }{XPPEDIT 18 0 "c=0.9" "6#/%\"cG-%&FloatG6$\"
\"*!\"\"" }{TEXT 223 8 "  , and " }{XPPEDIT 18 0 "d=4.5" "6#/%\"dG-%&F
loatG6$\"#X!\"\"" }{TEXT 223 3 "  ." }}{PARA 206 "" 0 "" {TEXT 222 49 
"Then the differential equations can be defined as" }}{EXCHG {PARA 
208 "> " 0 "" {MPLTEXT 1 229 8 "restart;" }}}{EXCHG {PARA 206 "> " 0 "
" {MPLTEXT 1 220 93 "de1 := diff(f(t), t) = -.16*f(t)+.08*f(t)*r(t);\n
de2 := diff(r(t), t) = 4.5*r(t)-.9*f(t)*r(t);" }}}{EXCHG {PARA 206 "" 
0 "" {TEXT 222 39 "And the initial conditions are given as" }}{PARA 
206 "> " 0 "" {MPLTEXT 1 220 33 "ic1 := f(0) = 4;\nic2 := r(0) = 4;" }
}}{EXCHG {PARA 206 "" 0 "" {TEXT 222 125 "The solution (obtained numer
ically - since an analytic solution is not known - using the default m
ethod) is then obtained via" }}{PARA 206 "> " 0 "" {MPLTEXT 1 220 70 "
ecosystem := dsolve(\{de1, de2, ic1, ic2\}, \{f(t), r(t)\}, type=numer
ic);" }{TEXT -1 0 "" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 222 70 "There a
re now basically two different ways of looking at the solution." }}
{PARA 206 "" 0 "" {TEXT 222 24 "1. As a list of numbers:" }}{PARA 206 
"> " 0 "" {MPLTEXT 1 220 49 "for dt from 0 by .1 to 1 do\n   ecosystem
(dt);\nod;" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 222 3 "or " }}{PARA 206 
"" 0 "" {TEXT 222 30 "2. As a plot of the functions " }{XPPEDIT 18 0 "
f" "6#%\"fG" }{TEXT 223 7 "   and " }{XPPEDIT 18 0 "r" "6#%\"rG" }
{TEXT 223 44 "  .\nTo this end we need to use the function " }
{HYPERLNK 17 "odeplot" 2 "odeplot" "" }{TEXT 223 10 " from the " }
{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT 223 34 " package (which also \+
contains the " }{HYPERLNK 17 "display" 2 "display" "" }{TEXT 223 59 " \+
command used to combine the two plots into a single plot)." }}{PARA 
206 "> " 0 "" {MPLTEXT 1 220 159 "with(plots):\nfoxes := odeplot(ecosy
stem, [[t,f(t)]], 0..20, color=red):\nrabbits := odeplot(ecosystem, [[
t,r(t)]], 0..20, color=brown):\ndisplay(foxes, rabbits);" }}}{EXCHG 
{PARA 208 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 207 "" 0 "" {TEXT 
221 12 "Assignment 5" }}{SECT 1 {PARA 207 "" 0 "" {TEXT 221 5 "Ex.1:" 
}}{PARA 206 "" 0 "" {TEXT 222 127 "a) Solve the refined free fall mode
l above with an arbitrary initial velocity (i.e., by not specifying an
 initial condition in " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT 
222 3 " )." }}{PARA 206 "" 0 "" {TEXT 222 15 "b) What is the " }{TEXT 
230 17 "terminal velocity" }{TEXT 222 94 " of a body under this model?
 (Note that this quantity is independent of the initial velocity.)" }}
{EXCHG {PARA 206 "" 0 "" {TEXT 222 48 "Hint: use the following command
s to assume that " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT 223 2 "  " }
{TEXT 222 5 " and " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 223 2 "  " }
{TEXT 222 25 " are positive quantities." }}{PARA 206 "> " 0 "" 
{MPLTEXT 1 220 25 "assume(k>0);\nassume(m>0);" }}}}{SECT 1 {PARA 207 "
" 0 "" {TEXT 221 5 "Ex.2:" }}{PARA 206 "" 0 "" {TEXT 222 70 "In an ele
mentary chemical reaction, single molecules of two reactants " }
{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT 223 2 "  " }{TEXT 222 5 " and " }
{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT 223 2 "  " }{TEXT 222 32 " form a m
olecule of the product " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 223 2 "  \+
" }{TEXT 222 22 " according to the rule" }}{PARA 206 "" 0 "" {TEXT 
222 5 "     " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT 223 2 "  " }{TEXT 
222 3 " + " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT 223 2 "  " }{TEXT 222 
4 " -> " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 223 2 "  " }{TEXT 222 2 "
. " }}{PARA 206 "" 0 "" {TEXT 222 4 "The " }{TEXT 230 18 "law of mass \+
action" }{TEXT 222 90 " states that the rate of reaction is proportion
al to the product of the concentrations of " }{XPPEDIT 18 0 "A" "6#%\"
AG" }{TEXT 223 2 "  " }{TEXT 222 5 " and " }{XPPEDIT 18 0 "B" "6#%\"BG
" }{TEXT 223 2 "  " }{TEXT 222 6 ":\n    " }{XPPEDIT 18 0 "d*[C]/dt = \+
k*[A]*[B];" "6#/*(%\"dG\"\"\"7#%\"CGF&%#dtG!\"\"*(%\"kGF&7#%\"AGF&7#%
\"BGF&" }{TEXT 223 2 "  " }{TEXT 222 2 ", " }}{PARA 206 "" 0 "" {TEXT 
222 45 "where [A], [B], and [C] are all functions of " }{XPPEDIT 18 0 
"t" "6#%\"tG" }{TEXT 223 2 "  " }{TEXT 222 6 ", and " }{XPPEDIT 18 0 "
k" "6#%\"kG" }{TEXT 223 2 "  " }{TEXT 222 59 " is a constant.\nLet's a
ssume the initial concentrations of " }{XPPEDIT 18 0 "A" "6#%\"AG" }
{TEXT 223 2 "  " }{TEXT 222 5 " and " }{XPPEDIT 18 0 "B" "6#%\"BG" }
{TEXT 223 2 "  " }{TEXT 222 22 " are equal, and write " }{XPPEDIT 18 
0 "[A](0) = [B](0);" "6#/-7#%\"AG6#\"\"!-7#%\"BG6#F(" }{TEXT 223 2 "  \+
" }{TEXT 222 3 " = " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT 223 2 "  " }
{TEXT 222 14 ".\nIf we write " }{XPPEDIT 18 0 "c(t) = [C](t)" "6#/-%\"
cG6#%\"tG-7#%\"CG6#F'" }{TEXT 223 2 "  " }{TEXT 222 26 " for the conce
ntration of " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 223 2 "  " }{TEXT 
222 9 " at time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 223 2 "  " }
{TEXT 222 28 ", then the concentration of " }{XPPEDIT 18 0 "A" "6#%\"A
G" }{TEXT 223 2 "  " }{TEXT 222 21 " (as well as that of " }{XPPEDIT 
18 0 "B" "6#%\"BG" }{TEXT 223 2 "  " }{TEXT 222 10 ") at time " }
{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 223 2 "  " }{TEXT 222 4 " is " }
{XPPEDIT 18 0 "[A](t) = a-[C](t)" "6#/-7#%\"AG6#%\"tG,&%\"aG\"\"\"-7#%
\"CG6#F(!\"\"" }{TEXT 223 2 "  " }{TEXT 222 2 ". " }}{PARA 206 "" 0 "
" {TEXT 222 69 "Therefore, in our case, the law of mass action can be \+
stated as \n    " }{XPPEDIT 18 0 "dc/dt = k*(a-c)^2;" "6#/*&%#dcG\"\"
\"%#dtG!\"\"*&%\"kGF&*$,&%\"aGF&%\"cGF(\"\"#F&" }{TEXT 223 2 "  " }
{TEXT 222 17 ".\na) Use Maple's " }{HYPERLNK 17 "dsolve" 2 "dsolve" "
" }{TEXT 222 17 " command to find " }{XPPEDIT 18 0 "c;" "6#%\"cG" }
{TEXT 223 2 "  " }{TEXT 222 23 ", the concentration of " }{XPPEDIT 18 
0 "C" "6#%\"CG" }{TEXT 223 2 "  " }{TEXT 222 19 ", as a function of " 
}{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 223 2 "  " }{TEXT 222 59 ". \nHint
: the reaction begins with an initial concentration " }{XPPEDIT 18 0 "
[C](0) = 0;" "6#/-7#%\"CG6#\"\"!F(" }{TEXT 223 2 "  " }{TEXT 222 1 ".
" }}{PARA 206 "" 0 "" {TEXT 222 57 "b) How does this expression simpli
fy if it is known that " }{XPPEDIT 18 0 "[C]=a/2" "6#/7#%\"CG*&%\"aG\"
\"\"\"\"#!\"\"" }{TEXT 223 2 "  " }{TEXT 222 18 " after 20 seconds?" }
}}{SECT 1 {PARA 207 "" 0 "" {TEXT 221 5 "Ex.3:" }}{PARA 206 "" 0 "" 
{TEXT 222 68 "In 1995 gray wolves were reintroduced in Yellowstone Nat
ional Park. " }}{PARA 206 "" 0 "" {TEXT 222 101 "The following is a mo
del of how the coyote, wolf and elk populations in Yellowstone might i
nteract.\n(" }{TEXT 256 5 "NOTE:" }{TEXT 222 36 " a number such as .4e
-1 means 0.04.)" }}{PARA 206 "" 0 "" {TEXT 222 3 "   " }{XPPEDIT 18 0 
"de/dt = .4e-1*e(t)-.3e-2*e(t)*c(t)-.85*e(t)*w(t);" "6#/*&%#deG\"\"\"%
#dtG!\"\",(*&-%&FloatG6$\"\"%!\"#F&-%\"eG6#%\"tGF&F&*(-F,6$\"\"$!\"$F&
-F16#F3F&-%\"cG6#F3F&F(*(-F,6$\"#&)F/F&-F16#F3F&-%\"wG6#F3F&F(" }
{TEXT 223 2 "  " }}{PARA 206 "" 0 "" {TEXT 222 3 "   " }{XPPEDIT 18 0 
"dc/dt = -.6e-1*c(t)+.1e-2*e(t)*c(t);" "6#/*&%#dcG\"\"\"%#dtG!\"\",&*&
-%&FloatG6$\"\"'!\"#F&-%\"cG6#%\"tGF&F(*(-F,6$F&!\"$F&-%\"eG6#F3F&-F16
#F3F&F&" }{TEXT 223 2 "  " }}{PARA 206 "" 0 "" {TEXT 222 3 "   " }
{XPPEDIT 18 0 "dw/dt = -.12*w(t)+.5e-2*e(t)*w(t);" "6#/*&%#dwG\"\"\"%#
dtG!\"\",&*&-%&FloatG6$\"#7!\"#F&-%\"wG6#%\"tGF&F(*(-F,6$\"\"&!\"$F&-%
\"eG6#F3F&-F16#F3F&F&" }{TEXT 223 3 "  ," }}{PARA 206 "" 0 "" {TEXT 
222 6 "where " }{XPPEDIT 18 0 "e(t)" "6#-%\"eG6#%\"tG" }{TEXT 223 4 " \+
 , " }{XPPEDIT 18 0 "c(t)" "6#-%\"cG6#%\"tG" }{TEXT 223 7 "   and " }
{XPPEDIT 18 0 "w(t)" "6#-%\"wG6#%\"tG" }{TEXT 223 56 "   denote the el
k, coyote, and wolf populations at time " }{XPPEDIT 18 0 "t" "6#%\"tG
" }{TEXT 223 58 "   (all populations are measured in thousands of anim
als)." }}{PARA 206 "" 0 "" {TEXT 222 47 "Note that this is also a Lotk
a-Volterra system." }}{PARA 206 "" 0 "" {TEXT 222 20 "At the initial t
ime " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT 223 62 "   (in 1995)
 there were 60000 elk, 2000 coyotes and 15 wolves." }}{PARA 206 "" 0 "
" {TEXT 222 79 "Use Maple to predict the development of this ecosystem
 over the next 300 years." }}{PARA 206 "" 0 "" {TEXT 222 155 "Hint: To
 get a nice plot containing all three species it helps to magnify the \+
coyote population by a factor of 10, and the wolf population by a fact
or 100." }}}}{PARA 211 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0 0" 8 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }