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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 
-1 60 "Using dsolve for solving first order differential equations." }
}{PARA 0 "" 0 "" {TEXT -1 31 "Michael Monagan, October 2005.\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "Newton's law of cooling  \nT(t) i
s the temperature of the body at time t\nTr   is the ambient (room) te
mperature of the surrounding medium\n k  is the cooling rate constant
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "de := diff(T(t),t) = k*
(Tr-T(t)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve( de, \+
T(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve( \{de,T(0
)=T0\}, T(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Tr := 20
; k := 0.1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sol := dsolv
e( \{de,T(0)=60\}, T(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "plot( rhs(sol), t=0..10, T=0..70 );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "initvals := [5,20,40,60];" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 56 "sols := [seq( dsolve(\{de,T(0)=T0\},T(t)), T0=initv
als )];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot( map(rhs,so
ls), t=0..20, T=0..70 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Logi
stic growth \ny(t) is the population at time t\nYm is the maximum sust
ainable population (carrying capacity)\nk = a Ym is the natural growth
 rate, i.e., a = k/Ym" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "re
start;\nde := diff(y(t),t)=a*y(t)*(Ym-y(t));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "dsolve( de, y(t) );" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "dsolve( \{de,y(0)=y0\}, y(t) );" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 21 "a := 0.02; Ym := 10; " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "sol := dsolve( \{de,y(0)=0.1\}, y(t) );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot( rhs(sol), t=0..60 );" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "iv := [0,0.1,1,5,8,10,12]
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "de;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 50 "sols := [seq( dsolve(\{de,y(0)=y0\},y(t)), \+
y0=iv) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot( map(rhs,
sols), t=0..50, y=-1..13, axes=frame, thickness=2 );" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 44 "Using dsolve to compute numerical solutions." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sol := dsolve( \{de,y(0)=1
\}, y(t), numeric );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol(
1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol(2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plots[odeplot](sol,[t,y(t)],t=0..60
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Logistic growth maximum sus
tainable population  Ym + b t  that is, increasing linearly in time. \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "de := diff(y(t),t) = a*y(t)*(Ym+b*t
-y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve( de, y(t)
 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dsolve( de, y(t) ) a
ssuming a>0,b>0,Ym>0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a \+
:= 0.02; Ym := 10; b := 0.1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "sol := dsolve( \{de,y(0)=1\}, y(t) );" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 30 "simplify( eval( sol, t=10 ) );" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 26 "plot( rhs(sol), t=0..60 );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sol := dsolve( \{de,y(0)=1\}, y(t),
 numeric );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plots[odeplo
t]( sol, [t,y(t)], t=0..60 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{MARK "27 0 0" 48 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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