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{TEXT 229 0 "" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG 
{PARA 229 "" 0 "" {TEXT 226 37 "What can you conclude from the graph?"
 }{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 58 "Could you identify \+
a curve where the velocity is constant?" }{TEXT 226 0 "" }}{PARA 229 "
" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 74 
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-%*LINESTYLEG6#F^^[l" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve \+
1" "Curve 2" }}{TEXT 229 0 "" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 226 0 
"" }}{PARA 229 "" 0 "" {TEXT 226 197 "The colored  line is a \"particu
lar solution\". As you can see for an \"initial condition\" v(0) = 49,
 the velocity remains constantly equall to 49 for all time. This is ca
lled an equilibrium solution." }{TEXT 226 0 "" }}{PARA 229 "" 0 "" 
{TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 74 "DEplot
(foeq, v(t),t=0..10,v=40..60,\{[0,58]\},arrows=THIN,title=`Figure 1`);
" }{MPLTEXT 1 227 0 "" }}{PARA 233 "" 1 "" {GLPLOT2D 400 400 400 
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-F$6%777$$\"\"!F^^[l$\"#eF^^[l7$$\"+++++]!#5$\"33+++]PN9dF-7$$\"+++++5
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+++++lFj^[l$\"3-u(pD!*y_9&F-7$$\"+++++qFj^[l$\"31yR+\\v$>7&F-7$$\"++++
+vFj^[l$\"3y')o*pT<35&F-7$$\"+++++!)Fj^[l$\"3g?>bHrq\"3&F-7$$\"+++++&)
Fj^[l$\"3mNf$[U:W1&F-7$$\"+++++!*Fj^[l$\"3;')*p>Cp([]F-7$$\"+++++&*Fj^
[l$\"3c<5)*))>hM]F-7$$Fi^[l!\")$\"3P6'zb(>!=-&F--%&STYLEG6#%%LINEG-Ff]
[l6#\"\"$-%&TITLEG6#Q)Figure~16\"-%+AXESLABELSG6$Q\"tFae[lQ%v(t)Fae[l-
F]][l6&F_][l$\"1#=t:!zg>!*F-Fie[l$\"2CK-)H+++?FJ-%*GRIDSTYLEG6#%,RECTA
NGULARG-%%VIEWG6$;$!\"&Fb][l$\"$0\"Fb][l;$\"2+++++++!R!#:$\"1-++++++h!
#9-%+PROJECTIONG6#F`][l-%*LINESTYLEG6#F^^[l" 1 2 2 0 10 1 2 6 1 4 2 
1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}{TEXT 229 0 "" }}}{EXCHG 
{PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 194 "In
 this case the colored curve represents another particular solution. W
e can see that for an initial velocity over v = 49 (v(0)=58), the velo
city  decreases and tends to v = 49 when time grows." }{TEXT 226 0 "" 
}}{PARA 229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" 
{MPLTEXT 1 227 74 "DEplot(foeq, v(t),t=0..10,v=40..60,\{[0,42]\},arrow
s=THIN,title=`Figure 1`);" }{MPLTEXT 1 227 0 "" }}{PARA 229 "" 0 "" 
{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 118 "Now, we see that for
 an \"initial velocity\" belows 49 the velocity grows and  tends to v \+
= 49 as time goes to infinity." }{TEXT 226 0 "" }}{PARA 229 "" 0 "" 
{TEXT 226 0 "" }}{PARA 233 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "
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"" {TEXT 226 21 "Defining the equation" }{TEXT 226 0 "" }}{PARA 229 ""
 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 32 "
moeq:=diff(p(t),t)=1/2*p(t)-450;" }{MPLTEXT 1 227 0 "" }}{PARA 231 "" 
1 "" {XPPMATH 20 "6#>I%moeqG6\"/-I%diffGI*protectedGF)6$-I\"pGF%6#I\"t
GF%F.,&F+#\"\"\"\"\"#!$]%F1" }{TEXT 20 0 "" }}}{EXCHG {PARA 230 "> " 0
 "" {MPLTEXT 1 227 67 "DEplot(moeq, p(t),t=0..5,p=600..1200,arrows=THI
N,title=`Figure 1`);" }{MPLTEXT 1 227 0 "" }}{PARA 229 "" 0 "" {TEXT 
226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 166 "Again, we can see an equil
ibrium solution at p = 900. This solution can be graphically obtained \+
including the initial point or condition inside the command DEplot as \+
" }{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 233 "" 1 
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AXESLABELSG6$Q\"tFhcxQ%p(t)Fhcx-%*GRIDSTYLEG6#%,RECTANGULARG-%%VIEWG6$
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LEG6#\"\"!" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}
{TEXT 229 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 77 "DEplot
(moeq, p(t),t=0..5,p=600..1200,\{[0,900]\},arrows=THIN,title=`Figure 1
`);" }{MPLTEXT 1 227 0 "" }}{PARA 229 "" 0 "" {TEXT 226 304 "We will s
ee next a \"diverging\" behavior from the equilibrium solution. It mea
ns if we start with an initial number of individuals in the population
 over p(0) = 900, the population will grow  without bound. Similarly, \+
for an initial condition below 900, the population will decay until it
 becomes extinct." }{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 0 "" 
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[D(F-7$$\"++++]FF_ex$\"3e-%3TimC-(F-7$$\"+++++IF_ex$\"31r0!yih\"fnF-7$
$\"++++]KF_ex$\"3_a=$p/)zgkF-7$$\"+++++NF_ex$\"3&\\B1V*yqAhF-7$$\"++++
]PF_ex$\"3n=fm[;gRdF-7$I*undefinedGI*protectedGFjhxFihxFhhxFhhxFhhxFhh
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$Q\"tFfixQ%p(t)Ffix-Fhbx6&Fjbx$\"1#=t:!zg>!*!#;F^jx$\"2CK-)H+++?Fhn-%*
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\"Fccx-%+PROJECTIONG6#F[cx-%*LINESTYLEG6#Ficx" 1 2 2 0 10 1 2 6 1 4 2 
1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}{TEXT 229 0 "" }}}{EXCHG 
{PARA 235 "" 0 "" {TEXT 231 32 "PROBLEM 21: CHEMICAL IN THE POND" }
{TEXT 231 0 "" }}}{EXCHG {PARA 236 "" 0 "" {TEXT 232 0 "" }}{PARA 229 
"" 0 "" {TEXT 226 21 "Defining the equation" }{TEXT 226 0 "" }}{PARA 
229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 
227 36 "chpeq:=diff(Am(t),t)=3-0.0003*Am(t);" }{MPLTEXT 1 227 0 "" }}
{PARA 231 "" 1 "" {XPPMATH 20 "6#>I&chpeqG6\"/-I%diffGI*protectedGF)6$
-I#AmGF%6#I\"tGF%F.,&\"\"$\"\"\"F+$!\"$!\"%" }{TEXT 20 0 "" }}}{EXCHG 
{PARA 230 "> " 0 "" {MPLTEXT 1 227 73 "DEplot(chpeq, Am(t),t=0..10000,
Am=0..20000,arrows=THIN,title=`Figure 1`);" }{MPLTEXT 1 227 0 "" }}
{PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 169 "Ag
ain, we can see an equilibrium solution at Am = 10000. This solution c
an be graphically obtained including the initial point or condition in
side the command DEplot as " }{TEXT 226 0 "" }}{PARA 229 "" 0 "" 
{TEXT 226 0 "" }}{PARA 233 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "
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$6%777$$\"\"!Fgbx$\"&+]\"Fgbx7$$\"+++++]!\"($\"3++](oHa.V\"F^x7$$\"+++
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Fccx$\"3+%Q&H0mSu7F^x7$$\"+++++DFccx$\"3D,&fL7%=O7F^x7$$\"+++++IFccx$
\"3)G$Go/dG.7F^x7$$\"+++++NFccx$\"3!*Qq+`(p\\<\"F^x7$$\"+++++SFccx$\"3
It%f+(zf]6F^x7$$\"+++++XFccx$\"3#Q\\Jq'4iH6F^x7$$F\\cxFccx$\"3C#>a-)ec
66F^x7$$\"+++++bFccx$\"3qmS!>rDg4\"F^x7$$\"+++++gFccx$\"3)>zca:]E3\"F^
x7$$\"+++++lFccx$\"3OMk2*pP62\"F^x7$$\"+++++qFccx$\"3])oM*G)G71\"F^x7$
$\"+++++vFccx$\"35N$\\#z,q_5F^x7$$\"+++++!)Fccx$\"3Ao?<(\\f`/\"F^x7$$
\"+++++&)Fccx$\"3,yT-48/R5F^x7$$\"+++++!*Fccx$\"3Wf$)=!>.O.\"F^x7$$\"+
++++&*Fccx$\"3oODdaD#*G5F^x7$$Fbcx!\"&$\"3j&yt7*Q*[-\"F^x-%&STYLEG6#%%
LINEG-F_bx6#\"\"$-%&TITLEG6#Q)Figure~16\"-%+AXESLABELSG6$Q\"tFjixQ&Am(
t)Fjix-Feax6&Fgax$\"1#=t:!zg>!*F5Fbjx$\"2CK-)H+++?F]bx-%*GRIDSTYLEG6#%
,RECTANGULARG-%%VIEWG6$;$!%+]Fjax$\"2)************\\5!#7;$!&++\"Fjax$
\"2'*************4#Fb[y-%+PROJECTIONG6#Fhax-%*LINESTYLEG6#Fgbx" 1 2 2 
0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}{TEXT 229 0 "
" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 237 "" 0 "" 
{TEXT 233 20 "ANALYTICAL SOLUTIONS" }{TEXT 233 0 "" }}}{EXCHG {PARA 
229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 
227 18 "dsolve(foeq,v(t));" }{MPLTEXT 1 227 0 "" }}{PARA 231 "" 1 "" 
{XPPMATH 20 "6#/-I\"vG6\"6#I\"tGF&,&\"#\\\"\"\"*&-I$expG6$I*protectedG
F0I(_syslibGF&6#,$F(#!\"\"\"\"&F+I$_C1GF&F+F+" }{TEXT 20 0 "" }}}
{EXCHG {PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 
154 "This expresion represents the \"general solution\". To obtain a p
articular solution, we need to include an initial value inside the abo
ve command as follows" }{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 
0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 28 "dsolve(\{foeq,v(
0)=58\},v(t));" }{MPLTEXT 1 227 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "
6#/-I\"vG6\"6#I\"tGF&,&\"#\\\"\"\"-I$expG6$I*protectedGF/I(_syslibGF&6
#,$F(#!\"\"\"\"&\"\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 229 "" 0 "" 
{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 131 "This expresion repre
sent a particular solution for the given initial value. Unfortunately,
 it is not a function v(t) yet. In fact, " }{TEXT 226 0 "" }}{PARA 
229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 
227 5 "v(3);" }{MPLTEXT 1 227 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "6#
-I\"vG6\"6#\"\"$" }{TEXT 20 0 "" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 
226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 133 "We can obtain a function r
eperesenting the above particular solution, and later plot that functi
on performing the following procedure" }{TEXT 226 0 "" }}{PARA 229 "" 
0 "" {TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 47 "First separate t
he rhs of the above expression." }{TEXT 226 0 "" }}{PARA 229 "" 0 "" 
{TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 41 "tempv:
= rhs(dsolve(\{foeq,v(0)=58\},v(t)));" }{MPLTEXT 1 227 0 "" }}{PARA 
231 "" 1 "" {XPPMATH 20 "6#>I&tempvG6\",&\"#\\\"\"\"-I$expG6$I*protect
edGF,I(_syslibGF%6#,$I\"tGF%#!\"\"\"\"&\"\"*" }{TEXT 20 0 "" }}}
{EXCHG {PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 
71 "Then, use that rhs to define a function v(t) using the command una
pply." }{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG 
{PARA 230 "> " 0 "" {MPLTEXT 1 227 21 "v1:=unapply(tempv,t);" }
{MPLTEXT 1 227 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "6#>I#v1G6\"f*6#I
\"tGF%F%6$I)operatorGF%I&arrowGF%F%,&\"#\\\"\"\"-I$expG6$I*protectedGF
2I(_syslibGF%6#,$9$#!\"\"\"\"&\"\"*F%F%F%" }{TEXT 20 0 "" }}}{EXCHG 
{PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 60 "To \+
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0 "" }}{PARA 229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" 
{MPLTEXT 1 227 6 "v1(3);" }{MPLTEXT 1 227 0 "" }}{PARA 231 "" 1 "" 
{XPPMATH 20 "6#,&\"#\\\"\"\"-I$expG6$I*protectedGF)I(_syslibG6\"6##!\"
$\"\"&\"\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 
227 14 "evalf(v1(3));;" }{MPLTEXT 1 227 0 "" }}{PARA 231 "" 1 "" 
{XPPMATH 20 "6#$\"+s/$RR&!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 229 "" 0
 "" {TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 44 "Now, we can make \+
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{TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 39 "plot(v
1(t), t=0..12, title=`Figura 1`);" }{MPLTEXT 1 227 0 "" }}{PARA 233 ""
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$Q\"tFe[lQ!Fe[l-%*GRIDSTYLEG6#%,RECTANGULARG-%%VIEWG6$;F`[l$\"21++++++
?\"!#:;$\"1#o>63z_'\\!#9$\"1#zSo2nj\"eFi\\l-%+PROJECTIONG6#F][l-%*LINE
STYLEG6#F)" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}
{TEXT 229 0 "" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 229 
"" 0 "" {TEXT 226 28 "Another particular solution " }{TEXT 226 0 "" }}
{PARA 229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" 
{MPLTEXT 1 227 28 "dsolve(\{foeq,v(0)=42\},v(t));" }{MPLTEXT 1 227 0 "
" }}{PARA 231 "" 1 "" {XPPMATH 20 "6#/-I\"vG6\"6#I\"tGF&,&\"#\\\"\"\"-
I$expG6$I*protectedGF/I(_syslibGF&6#,$F(#!\"\"\"\"&!\"(" }{TEXT 20 0 "
" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 42 "tempv2:= rhs(dsolve
(\{foeq,v(0)=42\},v(t)));" }{MPLTEXT 1 227 0 "" }}{PARA 231 "" 1 "" 
{XPPMATH 20 "6#>I'tempv2G6\",&\"#\\\"\"\"-I$expG6$I*protectedGF,I(_sys
libGF%6#,$I\"tGF%#!\"\"\"\"&!\"(" }{TEXT 20 0 "" }}}{EXCHG {PARA 230 "
> " 0 "" {MPLTEXT 1 227 22 "v2:=unapply(tempv2,t);" }{MPLTEXT 1 227 0 
"" }}{PARA 231 "" 1 "" {XPPMATH 20 "6#>I#v2G6\"f*6#I\"tGF%F%6$I)operat
orGF%I&arrowGF%F%,&\"#\\\"\"\"-I$expG6$I*protectedGF2I(_syslibGF%6#,$9
$#!\"\"\"\"&!\"(F%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 229 "" 0 "" 
{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 65 "Now, we can plot both
 particular solutions in the same t-v plane." }{TEXT 226 0 "" }}{PARA 
229 "" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 
227 48 "plot(\{v1(t), v2(t)\}, t=0..12, title=`Figura 2`);" }{MPLTEXT 
1 227 0 "" }}{PARA 233 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6*-%
'CURVESG6$7S7$$\"\"!F)$\"#eF)7$$\"3*)*******\\ech#!#=$\"3abF'3XGTv&!#;
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\"34+++X.u-5!#<$\"3B))ycRRXOcF27$$\"3%*****\\Fy:f7F@$\"3G<8`(4Q'*f&F27
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27$F`z$\"3S)3P^5)4L[F27$Fez$\"3P6upKu\\O[F2-Fjz6&F\\[lF`[lF][lF`[l-%&T
ITLEG6#Q)Figura~26\"-%+AXESLABELSG6$Q\"tF]elQ!F]el-%*GRIDSTYLEG6#%,REC
TANGULARG-%%VIEWG6$;F`[l$\"21++++++?\"!#:;$\"%oT!\"#$\"1/+++++Ke!#9-%+
PROJECTIONG6#F][l-%*LINESTYLEG6#F)" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 
45.0 1 0 "Curve 1" "Curve 2" }}{TEXT 229 0 "" }}}{EXCHG {PARA 238 "> "
 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 226 191 "The
se graphics are the same we previously drew using DEplot without obtai
ning the exact solution explicitly. In fact, they were obtained from n
umerical solutions of the differential equation." }{TEXT 226 0 "" }}
{PARA 229 "" 0 "" {TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 43 "Now
, let's consider the IVP with IC v(0)=0." }{TEXT 226 0 "" }}{PARA 229 
"" 0 "" {TEXT 226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 
27 "dsolve(\{foeq,v(0)=0\},v(t));" }{MPLTEXT 1 227 0 "" }}{PARA 231 ""
 1 "" {XPPMATH 20 "6#/-I\"vG6\"6#I\"tGF&,&\"#\\\"\"\"-I$expG6$I*protec
tedGF/I(_syslibGF&6#,$F(#!\"\"\"\"&!#\\" }{TEXT 20 0 "" }}}{EXCHG 
{PARA 230 "> " 0 "" {MPLTEXT 1 227 41 "tempv3:= rhs(dsolve(\{foeq,v(0)
=0\},v(t)));" }{MPLTEXT 1 227 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "6#
>I'tempv3G6\",&\"#\\\"\"\"-I$expG6$I*protectedGF,I(_syslibGF%6#,$I\"tG
F%#!\"\"\"\"&!#\\" }{TEXT 20 0 "" }}}{EXCHG {PARA 230 "> " 0 "" 
{MPLTEXT 1 227 22 "v3:=unapply(tempv3,t);" }{MPLTEXT 1 227 0 "" }}
{PARA 231 "" 1 "" {XPPMATH 20 "6#>I#v3G6\"f*6#I\"tGF%F%6$I)operatorGF%
I&arrowGF%F%,&\"#\\\"\"\"-I$expG6$I*protectedGF2I(_syslibGF%6#,$9$#!\"
\"\"\"&!#\\F%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 
226 0 "" }}{PARA 229 "" 0 "" {TEXT 226 50 "Plot of this particular sol
ution in the t-v plane." }{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 
226 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 227 44 "plot(\{v3(t)
,49\}, t=0..20, title=`Figura 2`);" }{MPLTEXT 1 227 0 "" }}{PARA 233 "
" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"\"!F)$
\"#\\F)7$$\"39LLLL3VfV!#=F*7$$\"3%pmm;H[D:)F/F*7$$\"3MLLLe0$=C\"!#<F*7
$$\"3iLLL3RBr;F6F*7$$\"3imm;zjf)4#F6F*7$$\"3ULL$e4;[\\#F6F*7$$\"3!)***
*\\i'y]!HF6F*7$$\"3oLL$ezs$HLF6F*7$$\"3=++]7iI_PF6F*7$$\"3Onmm;_M(=%F6
F*7$$\"3%QLL$3y_qXF6F*7$$\"3]+++]1!>+&F6F*7$$\"3J+++]Z/NaF6F*7$$\"3;++
+]$fC&eF6F*7$$\"3qLL$ez6:B'F6F*7$$\"3/nmm;=C#o'F6F*7$$\"3Mnmmm#pS1(F6F
*7$$\"3<++]i`A3vF6F*7$$\"3Rmmmm(y8!zF6F*7$$\"3K,+]i.tK$)F6F*7$$\"3!3++
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F*7$$\"3WLLezw5V5!#;F*7$$\"3/++v$Q#\\\"3\"FcpF*7$$\"3]LL$e\"*[H7\"FcpF
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$$\"3'pmmT6KU$=FcpF*7$$\"3eLLL`v&Q(=FcpF*7$$\"30++DOl5;>FcpF*7$$\"3A++
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\"39[fhIzRuRFcp7$Feo$\"3O_KC^kRZSFcp7$Fho$\"3ubqA@EV<TFcp7$F[p$\"3CF;u
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Fcp7$Fjq$\"3e$\\B#Rm\"ze%Fcp7$F]r$\"3W0rL(**49h%Fcp7$F`r$\"3%QLlofv]j%
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p7$F\\s$\"3XJ[!esB,r%Fcp7$F_s$\"3<\"HyQ%)R_s%Fcp7$Fbs$\"3\\$>OsVa%RZFc
p7$Fes$\"3Uc)[J?4Cv%Fcp7$Fhs$\"3!y]?%='*QjZFcp7$F[t$\"3Se\"pBVt\\x%Fcp
7$F^t$\"3cMe6S%*\\%y%Fcp7$Fat$\"3!p3wL/eQz%Fcp7$Fdt$\"3`\"z.QN/@![Fcp7
$Fgt$\"3x-_WpLD5[Fcp-Fjt6&F\\uF`uF]uF`u-%&TITLEG6#Q)Figura~26\"-%+AXES
LABELSG6$Q\"tF[_lQ!F[_l-%*GRIDSTYLEG6#%,RECTANGULARG-%%VIEWG6$;F`uFgt;
$!#)*!\"#$\"1-+++++)*\\!#9-%+PROJECTIONG6#F]u-%*LINESTYLEG6#F)" 1 2 2 
0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}{TEXT 229 0 "
" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 218 0 "" }}{PARA 239 "" 0 "" 
{TEXT 218 72 "Now, we would like to find the time \"t\" when the objec
t hits the ground." }{TEXT 218 0 "" }}{PARA 239 "" 0 "" {TEXT 218 0 ""
 }}{PARA 239 "" 0 "" {TEXT 218 112 "First, we will integrate the veloc
ity solution with respect to time to obtain the position function at a
ny time." }{TEXT 218 0 "" }}{PARA 239 "" 0 "" {TEXT 218 0 "" }}}
{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 23 "aux:=int(v3(t),t=0..t);" }
{MPLTEXT 1 0 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "6#>I$auxG6\",(I\"tG
F%\"#\\-I$expG6$I*protectedGF,I(_syslibGF%6#,$F'#!\"\"\"\"&\"$X#!$X#\"
\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 18 "x:
=unapply(aux,t);" }{MPLTEXT 1 0 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "
6#>I\"xG6\"f*6#I\"tGF%F%6$I)operatorGF%I&arrowGF%F%,(9$\"#\\-I$expG6$I
*protectedGF2I(_syslibGF%6#,$F-#!\"\"\"\"&\"$X#!$X#\"\"\"F%F%F%" }
{TEXT 20 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 218 0 "" }}{PARA 239 "
" 0 "" {TEXT 218 61 "Here, we will need to solve the nonlinear algebra
ic equation:" }{TEXT 218 0 "" }}{PARA 239 "" 0 "" {TEXT 218 39 "      \+
          300=x(t). (solve for t)" }{TEXT 218 0 "" }}{PARA 239 "" 0 ""
 {TEXT 218 0 "" }}}{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 24 "aux2:=s
olve(x(t)=300,t);" }{MPLTEXT 1 0 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 
"6#>I%aux2G6\"6$,&-I)LambertWG6$I*protectedGF+I(_syslibGF%6#,$-I$expGF
*6##!$4\"\"#\\!\"\"\"\"&#\"$X&F4\"\"\",&-F)6$F5F.F6F7F9" }{TEXT 20 0 "
" }}}{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 12 "evalf(aux2);" }
{MPLTEXT 1 0 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "6$$\"+q$e60\"!\")$!
*wF!=iF%" }{TEXT 20 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 218 0 "" }}
{PARA 239 "" 0 "" {TEXT 218 9 "PROBLEM 9" }{TEXT 218 0 "" }}{PARA 239 
"" 0 "" {TEXT 218 23 "Time to reach 98% of 49" }{TEXT 218 0 "" }}
{PARA 239 "" 0 "" {TEXT 218 53 "We need to solve the nonlinear equatio
n for time \"t\"." }{TEXT 218 0 "" }}{PARA 239 "" 0 "" {TEXT 218 0 "" 
}}}{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 11 "vn:=49*.98;" }{MPLTEXT 
1 0 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "6#>I#vnG6\"$\"%-[!\"#" }
{TEXT 20 0 "" }}}{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 22 "tn:=solve
(v3(t)=vn,t);" }{MPLTEXT 1 0 0 "" }}{PARA 231 "" 1 "" {XPPMATH 20 "6#>
I#tnG6\"$\"+.:,c>!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 239 "" 0 "" 
{TEXT 218 0 "" }}{PARA 239 "" 0 "" {TEXT 218 30 "Space travelled durin
g time tn" }{TEXT 218 0 "" }}{PARA 239 "" 0 "" {TEXT 218 0 "" }}}
{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 6 "x(tn);" }{MPLTEXT 1 0 0 "" 
}}{PARA 231 "" 1 "" {XPPMATH 20 "6#$\"+ljX$=(!\"(" }{TEXT 20 0 "" }}}
{PARA 240 "" 0 "" {TEXT 234 0 "" }}{PARA 241 "" 0 "" {TEXT -1 0 "" }}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 
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