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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Problem 13 on page 99 of \+
Nagle & Saff asks for a prediction of future population using the logi
stic model " }{TEXT -1 34 "given by the differential equation" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "diffeq:= diff(P(t),t)=a*P(t)
-b*P(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'diffeqG/-%%diffG6$-%
\"PG6#%\"tGF,,&*&%\"aG\"\"\"F)F0F0*&%\"bGF0F)\"\"#!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 26 "with the initial condition" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "init:= P(0)=1000;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%%initG/-%\"PG6#\"\"!\"%+5" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 104 "Maple's dsolve command yields the following general s
olution (converted into an arrow-defined function):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 58 "P:=unapply(simplify(rhs(dsolve(\{diffeq, \+
init\}, P(t)))),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG:6#%\"tG6
\"6$%)operatorG%&arrowGF(,$*&%\"aG\"\"\",(%\"bG\"%+5*&-%$expG6#,$*&F.F
/9$F/!\"\"F/F.F/F/*&F4F/F1F/!%+5F:F2F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 148 "Notice that this result agrees with formula 15 on page 9
5 of the textbook (derived by separating variables and integrating usi
ng partial fractions)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "To dete
rmine the values of the parameters " }{XPPEDIT 18 0 "a" "I\"aG6\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 56 ", use t
he given information that the population at time " }{XPPEDIT 18 0 "t=7
" "/%\"tG\"\"(" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "3000" "\"%+I" }
{TEXT -1 29 ", and the population at time " }{XPPEDIT 18 0 "t=14" "/%
\"tG\"#9" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "5000" "\"%+]" }{TEXT -1 
89 ". This gives two equations for two unknowns. Here is what Maple's \+
solve command produces:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "
solve(\{P(7)=3000, P(14)=5000\}, \{a,b\}); assign(\");" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<$/%\"bG,$-%#lnG6#\"\"&#\"\"\"\"&+?%/%\"aG,$F'#F,
\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Thus, the final solution
 to the differential equation is the following:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "simplify(P(t));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*$,&\"\"\"F&)\"\"&,&F&F&%\"tG#!\"\"\"\"(F&F,\"%+g" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 109 "Because of the decaying exponential func
tion in the denominator, it is clear that the limiting population as \+
" }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 22 " tends to infinity is " 
}{XPPEDIT 18 0 "6000" "\"%+g" }{TEXT -1 22 ". Maple confirms this:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(P(t), t=infinity);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%+g" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 23 "The population at time " }{XPPEDIT 18 0 "t=21" "/%\"tG\"#
@" }{TEXT -1 17 " is approximately" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "round(P(21)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%
pd" }}}}{MARK "14 1 0" 0 }{VIEWOPTS 1 1 0 2 1 1805 }