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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 19 "An\341lise Qualitativa
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Vamos
 mais uma vez considerar a equa\347\343o log\355stica" }}{PARA 256 "" 
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"" 0 "" {TEXT -1 47 "mas encarando de um ponto de vista qualitativo." 
}}{PARA 0 "" 0 "" {TEXT -1 93 "O desenho unidimensional no eixo y, com
 setas representando derivadas n\343o sai f\341cil no Maple." }}{PARA 
0 "" 0 "" {TEXT -1 43 "Por outro lado, \351 f\341cil fazer o gr\341fic
o de " }{XPPEDIT 18 0 "f(y)=y*(1-y)" "6#/-%\"fG6#%\"yG*&F'\"\"\",&F)F)
F'!\"\"F)" }{TEXT -1 80 ", uma fun\347\343o real de uma vari\341vel re
al (n\343o h\341 depend\352ncia expl\355cita no tempo!)." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with
(plots):with(DETools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f
:=y*(1-y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&%\"yG\"\"\",&F'F
'F&!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(f,y=-1..
2);" }}{PARA 13 "" 1 "" {GLPLOT2D 426 144 144 {PLOTDATA 2 "6%-%'CURVES
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "Vamos tomar cuidado para n\343o c
onfundir as coisas. O gr\341fico acima nada tem a ver com o tempo. Por
 outro lado, h\341 de fato uma EDO que se relaciona com o gr\341fico, \+
a saber via y'(t)=f(y(t)). " }}{PARA 0 "" 0 "" {TEXT -1 90 "Note que a
s \372nicas solu\347\365es constantes y(t)=c da EDO em quest\343o s
\343o as ra\355zes da fun\347\343o f." }}{PARA 0 "" 0 "" {TEXT -1 57 "
De fato, substituindo y(t)=c para todo t na equa\347\343o temos" }}
{PARA 257 "" 0 "" {TEXT -1 27 "(c)'=f(c), ou seja, 0=f(c)." }}{PARA 0 
"" 0 "" {TEXT -1 39 "Pontos onde f se anula s\343o chamados de " }
{TEXT 273 11 "equil\355brios" }{TEXT -1 68 " da EDO, e dividem o eixo \+
y em regi\365es relevantes. No caso s\343o tr\352s:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "(I) y<0" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "como f<0 nessa regi
\343o, vemos que uma part\355cula com velocidade y'=f(y) iria sempre p
ara a esquerda e ter\355amos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
258 "" 0 "" {XPPEDIT 18 0 "limit(y(t),t=infinity)=-infinity" "6#/-%&li
mitG6$-%\"yG6#%\"tG/F*%)infinityG,$F,!\"\"" }{TEXT -1 3 ".  " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Bem, na verdade
 essa \351 uma meia-verdade. A sutileza, como j\341 vimos ao estudar o
 " }{TEXT 274 33 "teorema de exist\352ncia e unicidade" }{TEXT -1 93 "
, \351 que, mesmo quando f satisfaz a hip\363tese do teorema em todo e
spa\347o, a solu\347\343o de y'=f(y), " }{TEXT 275 8 "y(t0)=y0" }
{TEXT -1 142 ", pode n\343o estar definida para tempos arbitrariamente
 grandes. Nesse caso,  o que pode ocorrer \351 que f se torne ilimitad
a j\341 para algum valor " }{TEXT 276 7 "finito " }{TEXT -1 99 "t=a. N
esse caso, n\343o temos o direito de tomar o limite para t infinito e \+
devemos escrever em vez:  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
259 "" 0 "" {XPPEDIT 18 0 "limit(y(t),t = a,left) = -infinity;" "6#/-%
&limitG6%-%\"yG6#%\"tG/F*%\"aG%%leftG,$%)infinityG!\"\"" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "(II)
 0<y<1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
142 "aqui f>0 e a velocidade de uma part\355cula a levaria sempre para
 sua direita. Como n\343o podemos nunca atingir exatamente 1 (unicidad
e), vemos que" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{XPPEDIT 18 0 "limit(y(t),t=infinity)=1" "6#/-%&limitG6$-%\"yG6#%\"tG/
F*%)infinityG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 9 "(III) 1<y" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 75 "aqui, como f<0, as velocidades negati
vas na regi\343o nos levam de volta a 1 e" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "limit(y(t),t=infinity)=1" "6#/
-%&limitG6$-%\"yG6#%\"tG/F*%)infinityG\"\"\"" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 27 "Classificando \+
equili\355brios:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 203 "(a) y=0: note que se estivermos \340 direita, f \351 pos
itiva e portanto somos afastados de 0. Se for \340 esquerda, f ser\341
 negativa e novamente nos afastamos. O gr\341fico abaixo de algumas so
lu\347\365es ilustra o fato:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 194 "# edo\nlogistica:=diff(y(t),t)=y(t)*(1-y(t));\n# cond. iniciais
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red,blue]:\nestilo:=arrows=none,axes=boxed,linecolor=cl,thickness=1:" 
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GF,*&F)\"\"\",&F.F.F)!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 48 "DEplot(logistica,y(t),t=0..2,y=-1..2,ci,estilo);" }}{PARA 13 "" 
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1 2 0 1 10 1 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "De \+
fato, tanto a curva vermelha quanto a azul se afastam da solu\347\343o
 constante verde y=0 a medida que o tempo cresce." }}{PARA 0 "" 0 "" 
{TEXT -1 46 "Devido a esse comportamento, chamamos 0 de um " }{TEXT 
285 19 "equil\355brio inst\341vel" }{TEXT -1 16 " e tamb\351m de um " 
}{TEXT 284 8 "repulsor" }{TEXT -1 4 " ou " }{TEXT 283 5 "fonte" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 3 "(b)" }}{PARA 0 "" 0 "" {TEXT -1 182 "y=1: agora se estiver
mos \340 direita, f<0 e somos trazidos de volta em dire\347\343o a 1, \+
enquanto \340 esquerda, f>0 e novamente somos atra\355dos para 1. Nova
mente ilustrado pelo gr\341fico abaixo:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 133 "Vamos esbo\347ar um gr\341fico com \+
as solu\347\365es constantes y=0 e y=1, bem como solu\347\365es t\355p
icas em cada uma das tr\352s regi\365es intermedi\341rias." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "# edo\nlogistica:=diff(y(t),t)=y(t
)*(1-y(t));\n# cond. iniciais\nci:=[y(0)=0,y(0)=1,\ny(0)=1.5,y(0)=0.5]
:\n# cores\ncl:=[green,green,red,blue]:\nestilo:=arrows=none,axes=boxe
d,linecolor=cl,thickness=1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*logi
sticaG/-%%diffG6$-%\"yG6#%\"tGF,*&F)\"\"\",&F.F.F)!\"\"F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DEplot(logistica,y(t),t=0..2,y=-1..
2,ci,estilo);" }}{PARA 13 "" 1 "" {GLPLOT2D 838 164 164 {PLOTDATA 2 "6
*-%'CURVESG6&7S7$$\"\"!F)F(7$$\"+nmmmT!#6F(7$$\"+MLLL$)F-F(7$$\"++++]7
!#5F(7$$\"+nmmm;F4F(7$$\"+MLL$3#F4F(7$$\"+,+++DF4F(7$$\"+omm;HF4F(7$$
\"+NLLLLF4F(7$$\"+-++]PF4F(7$$\"+pmmmTF4F(7$$\"+OLL$e%F4F(7$$\"+.+++]F
4F(7$$\"+qmm;aF4F(7$$\"+PLLLeF4F(7$$\"+/++]iF4F(7$$\"+rmmmmF4F(7$$\"+Q
LL$3(F4F(7$$\"+0+++vF4F(7$$\"+smm;zF4F(7$$\"+RLLL$)F4F(7$$\"+1++]()F4F
(7$$\"+tmmm\"*F4F(7$$\"+SLL$e*F4F(7$$\"+,+++5!\"*F(7$$\"+ommT5F^pF(7$$
\"+NLL$3\"F^pF(7$$\"+-++D6F^pF(7$$\"+pmmm6F^pF(7$$\"+OLL37F^pF(7$$\"+.
++]7F^pF(7$$\"+qmm\"H\"F^pF(7$$\"+PLLL8F^pF(7$$\"+/++v8F^pF(7$$\"+rmm;
9F^pF(7$$\"+QLLe9F^pF(7$$\"+0+++:F^pF(7$$\"+smmT:F^pF(7$$\"+RLL$e\"F^p
F(7$$\"+1++D;F^pF(7$$\"+tmmm;F^pF(7$$\"+SLL3<F^pF(7$$\"+2++]<F^pF(7$$
\"+umm\"z\"F^pF(7$$\"+TLLL=F^pF(7$$\"+3++v=F^pF(7$$\"+vmm;>F^pF(7$$\"+
ULLe>F^pF(7$$\"\"#F)F(-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-%&STYLEG6#%
%LINEG-%*THICKNESSG6#\"\"\"-F$6&7S7$F($FeuF)7$F+$\"2y***************!#
<7$F/F\\v7$F2$\"3W+++++++5F^v7$F6Fav7$F9Fju7$F<$\"2))***************F^
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Z\"F^v7$F/$\"3uMR@![QBW\"F^v7$F2$\"3<6!)*o(Gw;9F^v7$F6$\"3O=_6&*)oIR\"
F^v7$F9$\"3m#yC!G_2r8F^v7$F<$\"3!*4RIw<i]8F^v7$F?$\"3C>4*3.o:L\"F^v7$F
B$\"3ae%H+d!z88F^v7$FE$\"3_;@')3\"zrH\"F^v7$FH$\"3]yN^@`j\"G\"F^v7$FK$
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>\"F^v7$Fjn$\"3!=g\">h4)o=\"F^v7$F]o$\"3)\\?NVM(*y<\"F^v7$F`o$\"3_k:+y
sSp6F^v7$Fco$\"3qNrbV!y8;\"F^v7$Ffo$\"3_qmK=%zP:\"F^v7$Fio$\"3!H)eZkOe
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S3\"F^v7$Far$\"35%)***f%HN!3\"F^v7$Fdr$\"3e$R\\Iy@o2\"F^v7$Fgr$\"3)=zA
4RcM2\"F^v7$Fjr$\"3S_&H^/[-2\"F^v7$F]s$\"3K9&pKd)=n5F^v7$F`s$\"3y')[i*
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\\5F^v7$Fet$\"33r0+XGCZ5F^v-Fht6&FjtF[uF(F(F^uFbu-F$6&7S7$F($\"/++++++
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\"3\\Z)*Q(*f0CdFicl7$FB$\"3x^3$z/<d#eFicl7$FE$\"3!yP&[O*ym#fFicl7$FH$
\"3Ya@O@G'o-'Ficl7$FK$\"3QBk@DE>EhFicl7$FN$\"3GJ*\\].&fCiFicl7$FQ$\"3J
W4@-&**>K'Ficl7$FT$\"3cKURU#Q$=kFicl7$FW$\"3P$3TcBYN^'Ficl7$FZ$\"3ik<O
c9c2mFicl7$Fgn$\"3_ps&)[jK+nFicl7$Fjn$\"3yR`H(e'y\"z'Ficl7$F]o$\"39-v$
Hc\"*=)oFicl7$F`o$\"3Aa:dTXfqpFicl7$Fco$\"3=S:^jE&y0(Ficl7$Ffo$\"3=mxf
VpiVrFicl7$Fio$\"3G)y'>rA)yA(Ficl7$F\\p$\"3&QD\"z5ue5tFicl7$F`p$\"3q3v
=+]r\"R(Ficl7$Fcp$\"3?B/L_:CruFicl7$Ffp$\"3s?7pau9\\vFicl7$Fip$\"3jQz#
e!pTDwFicl7$F\\q$\"3L[Y()*=O+q(Ficl7$F_q$\"3aspPUb*Hx(Ficl7$Fbq$\"3G&Q
A'e()GWyFicl7$Feq$\"3t8eqaC\"R\"zFicl7$Fhq$\"3WS'Hv1m=)zFicl7$F[r$\"3A
,P![&=:[!)Ficl7$F^r$\"3Qz_/&*[x7\")Ficl7$Far$\"3E>\"zv3Vd<)Ficl7$Fdr$
\"3:'p(45q1P#)Ficl7$Fgr$\"3y0&3t#)enH)Ficl7$Fjr$\"3-Wc())RL[N)Ficl7$F]
s$\"3U59zjtI6%)Ficl7$F`s$\"37'*R&o)))>m%)Ficl7$Fcs$\"3R)\\n'ew_>&)Ficl
7$Ffs$\"33')=k&*[Jr&)Ficl7$Fis$\"31LI**RLe@')Ficl7$F\\t$\"3!fpU(fsNq')
Ficl7$F_t$\"3Ex95<Am<()Ficl7$Fbt$\"3#G%3bRU_j()Ficl7$Fet$\"3-gRf70(z!)
)Ficl-Fht6&FjtF(F(F[uF^uFbu-%%VIEWG6$;F(Fet;$!\"\"F)FetFbu-%*AXESSTYLE
G6#%$BOXG-%+AXESLABELSG6$Q\"t6\"Q%y(t)Fh]m" 1 2 0 1 10 1 2 9 1 2 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Agora dizemos que y=1 \351 \+
um " }{TEXT 288 18 "equil\355brio est\341vel" }{TEXT -1 19 " e tamb
\351m que \351 um " }{TEXT 287 7 "atrator" }{TEXT -1 8 ", ou um " }
{TEXT 286 9 "sumidouro" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 101 "Dito isso, vamos fazer uma figura ag
ora com as duas solu\347\365es constantes e uma solu\347\343o em cada \+
regi\343o." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "# cond. inic
iais\nci:=[y(0)=0,y(0)=1,\ny(0)=-.5,y(0)=.5,y(0)=1.5]:\n# cores\ncl:=[
green,green,red,blue,black]:\n# estilo\nestilo:=arrows=none,axes=boxed
,linecolor=cl,thickness=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
65 "g1:=DEplot(logistica,y(t),t=0..2,y=-5..2,ci,estilo):\ndisplay(g1);
" }}{PARA 13 "" 1 "" {GLPLOT2D 750 272 272 {PLOTDATA 2 "6*-%'CURVESG6&
7S7$$\"\"!F)F(7$$\"+nmmmT!#6F(7$$\"+MLLL$)F-F(7$$\"++++]7!#5F(7$$\"+nm
mm;F4F(7$$\"+MLL$3#F4F(7$$\"+,+++DF4F(7$$\"+omm;HF4F(7$$\"+NLLLLF4F(7$
$\"+-++]PF4F(7$$\"+pmmmTF4F(7$$\"+OLL$e%F4F(7$$\"+.+++]F4F(7$$\"+qmm;a
F4F(7$$\"+PLLLeF4F(7$$\"+/++]iF4F(7$$\"+rmmmmF4F(7$$\"+QLL$3(F4F(7$$\"
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F4F(7$$\"+SLL$e*F4F(7$$\"+,+++5!\"*F(7$$\"+ommT5F^pF(7$$\"+NLL$3\"F^pF
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>h4)o=\"F^v7$F]o$\"3)\\?NVM(*y<\"F^v7$F`o$\"3_k:+ysSp6F^v7$Fco$\"3qNrb
V!y8;\"F^v7$Ffo$\"3_qmK=%zP:\"F^v7$Fio$\"3!H)eZkOeY6F^v7$F\\p$\"3X:(31
Hl(R6F^v7$F`p$\"36\\OF/5IL6F^v7$Fcp$\"3NUwad\"pr7\"F^v7$Ffp$\"3%)4xV*f
\\87\"F^v7$Fip$\"3oF4)enBe6\"F^v7$F\\q$\"3WEnY3Td56F^v7$F_q$\"3h%=E!y
\\e06F^v7$Fbq$\"3DOM\"oZT35\"F^v7$Feq$\"3!zsNCqHj4\"F^v7$Fhq$\"3?An%yl
O?4\"F^v7$F[r$\"3w%3#QW-&z3\"F^v7$F^r$\"3%*zWnc\"fS3\"F^v7$Far$\"35%)*
**f%HN!3\"F^v7$Fdr$\"3e$R\\Iy@o2\"F^v7$Fgr$\"3)=zA4RcM2\"F^v7$Fjr$\"3S
_&H^/[-2\"F^v7$F]s$\"3K9&pKd)=n5F^v7$F`s$\"3y')[i*GqU1\"F^v7$Fcs$\"3cE
0nZg[h5F^v7$Ffs$\"3A`uzE\"H)e5F^v7$Fis$\"3MJ.SYJHc5F^v7$F\\t$\"3Stu(e1
sQ0\"F^v7$F_t$\"3cP3j%=g:0\"F^v7$Fbt$\"3#)z**\\O@N\\5F^v7$Fet$\"33r0+X
GCZ5F^v-Fht6&FjtF(F(F(F^uFbu-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6$Q\"t6
\"Q%y(t)F]bm-%%VIEWG6$;F(Fet;$!\"&F)Fet" 1 2 0 1 10 0 2 9 1 2 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Vimos na aula te
\363rica que, por unicidade, solu\347\365es est\343o sempre confinadas
 a cada uma das regi\365es entre as constantes." }}{PARA 0 "" 0 "" 
{TEXT -1 260 "em particular, solu\347\365es como a curva azul acima se
r\343o sempre limitadas, pois nunca poder\343o sequer tocar nas retas \+
verde. J\341 as curvas vermelha e preta n\343o t\352m essa propriedade
. De fato, vamos calcular a express\343o em forma fechada para a solu
\347\343o vermelha acima:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "dsolve([logistica,y(0)=-.5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%\"yG6#%\"tG,$*&\"\"\"F*,&F*!\"\"*&\"\"$F*-%$expG6#,$F'F,F*F*F,F," }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solverm:=rhs(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%(solvermG,$*&\"\"\"F',&F'!\"\"*&\"\"$F'-%$
expG6#,$%\"tGF)F'F'F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "
g2:=plot(solverm,t=0..2,color=magenta,view=[0..2,-5..5]):\ndisplay(g2)
;" }}{PARA 13 "" 1 "" {GLPLOT2D 668 232 232 {PLOTDATA 2 "6%-%'CURVESG6
$7jp7$$\"\"!F)$!3++++++++]!#=7$$\"39LLLL3VfV!#>$!3UId0')Q!=M&F,7$$\"3'
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w%>'3#F^v7$$\"3%*******p(G**y\"F^v$\"3i=F\\z9n.?F^v7$$\"3lmm;9@BM=F^v$
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v$\"3%*=Dp.yi!z\"F^v7$$\"3/++v.Uac>F^v$\"3'*RS_CgAO<F^v7$Fet$\"3?3Myi#
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%y(t)Fg_o-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fet;$!\"&F)\"\"&" 1 2 0 1 
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urve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 117 "Que parece contradizer que solu\347\365es n\343o cruzam \+
regi\365es. Afinal a curva rosa aparece na regi\343o inferior e na sup
erior!" }}{PARA 0 "" 0 "" {TEXT -1 49 "Ainda, calculando o limite a pa
rtir da express\343o:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "li
mit(solverm,t=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "!?" }}{PARA 0 "" 0 "" {TEXT -1 
114 "Por outro lado, esse limite vem justamente da parte da curva rosa
 acima de y=1. O que ocorre \351 que o numerador de " }{XPPEDIT 18 0 "
-1/(-1+3*exp(-t));" "6#,$*&\"\"\"F%,&F%!\"\"*&\"\"$F%-%$expG6#,$%\"tGF
'F%F%F'F'" }{TEXT -1 38 " se anula em tempo finito. A saber, em" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(-1+3*exp(-t)=0,t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+
*G7')4\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Ou seja, essa exp
ress\343o em forma fechada " }{TEXT 265 10 "n\343o define" }{TEXT -1 
30 " uma solu\347\343o deriv\341vel da EDO " }{TEXT 266 11 "para todo \+
t" }{TEXT -1 159 ", somente para t<ln(3) (j\341 que demos uma cond. in
icial em t=0). Esse \351 o caso em que y se torna ilimitada j\341 em t
empo finito, e na verdade, o limite infinito \351" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "limit(y(t),t=ln(3))=-infinity" "6#/-%&limitG6$-%\"yG6#%
\"tG/F*-%#lnG6#\"\"$,$%)infinityG!\"\"" }{TEXT -1 7 " e n\343o " }
{XPPEDIT 18 0 "limit(y(t),t=infinity)=-infinity" "6#/-%&limitG6$-%\"yG
6#%\"tG/F*%)infinityG,$F,!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "Isso explica o aparente \+
paradoxo. A regi\343o n\343o cruza a descontinuidade y=ln(3). Ela s
\363 faz sentido em y<3 ou y>3. Como estamos pedindo y(0)=-.5, trata-s
e de y<3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 31 "Ok, chega de equa\347\343o log\355stica." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Um outro exemplo de edo d
a forma" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
7 "y'=f(y)" }}{PARA 0 "" 0 "" {TEXT -1 10 "agora com " }{XPPEDIT 18 0 
"f(y)=y^2-y^4" "6#/-%\"fG6#%\"yG,&*$F'\"\"#\"\"\"*$F'\"\"%!\"\"" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f:=y^2-y^4
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iffG6$-%\"yG6#%\"tGF,,&*$)F)\"\"#\"\"\"F1*$)F)\"\"%F1!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# gr\341fico de f\nplot(f,y=-2..2,v
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{PARA 0 "" 0 "" {TEXT -1 29 "Vemos agora tr\352s equil\355brios:" }}
{PARA 0 "" 0 "" {TEXT -1 21 "(a) y=-1: equil\355brio " }{TEXT 268 8 "i
nst\341vel" }{TEXT -1 5 ", um " }{TEXT 267 8 "repulsor" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) y= 0: nem atrator nem repulsor, m
as uma " }{TEXT 269 4 "sela" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 20 "(c) y=1: equil\355brio " }{TEXT 271 7 "est\341vel" }{TEXT -1 5 
", um " }{TEXT 270 7 "atrator" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Note que podemos relacio
nar a derivada de f (n\343o confundir com a derivada de y(t) na edo!!)
 no equil\355brio e que tipo de equil\355brio temos:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Se em y* tivermos f(y*)
=0 f '(y*)>0, temos um repulsor." }}{PARA 0 "" 0 "" {TEXT -1 55 "Se em
 y* tivermos f(y*)=0 f '(y*)<0, temos um atyrator." }}{PARA 0 "" 0 "" 
{TEXT -1 105 "Pontos de sela tem que ter derivada (de f!!) nula, mas n
em sempre isso garante que seja sela (considere  " }{XPPEDIT 18 0 "f=y
^3" "6#/%\"fG*$%\"yG\"\"$" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "f=-y^3" "
6#/%\"fG,$*$%\"yG\"\"$!\"\"" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Novamente desenhos:" }}}{EXCHG 
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ck,black,black,\nred,magenta,blue,cyan]:\n# estilo\nestilo:=arrows=non
e,axes=boxed,linecolor=cl,thickness=1:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "g2:=DEplot(edo,y(t),t=0..2,y=-3..2,ci,estilo):" }}}
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{PARA 0 "" 0 "" {TEXT -1 214 "E podemos verificar tamb\351m no gr\341f
ico acima os tipos de equil\355brio. Note que a curva vermelha tamb
\351m parece ser ilimitada em tempo finito. Achar tal tempo agora \351
 mais complicado, j\341 que a solu\347\343o da EDO \351 dada por" }}
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{PARA 0 "" 0 "" {TEXT -1 200 "Vemos que \351 uma situa\347\343o pareci
da com o exemplo anterior, com o agravante que o Maple se enrola todo \+
para resolver a raiz na express\343o acima. Note que at\351 mesmo a co
ndi\347\343o inicial (-1.5) aparece como " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "subs(t=0,sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
&-%$expG6#-%'RootOfG6#,6*(\"\"$\"\"\"%#_ZGF--F%6#F.F-F-*&F,F-F.F-!\"\"
\"\"#F2*(F,F-F/F--%#lnG6#,&F/F-F3F2F-F2*&F,F-F5F-F-*&\"\"%F-F/F-F2*(F,
F-F/F--F66#F3F-F-*(F,F-F/F--F66##\"\"&F3F-F-*&F,F-F=F-F2*&F,F-F@F-F2F-
F-F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#^$$!+++++:!\"*$!+3Q.^?!#>" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 49 "que, apesar de ter uma parte imagin\341ria da o
rdem " }{XPPEDIT 18 0 "10^(-9);" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 12 " \+
nem real \351." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 97 "Em todos exemplos at\351 agora, as solu\347\365es s\343o \+
sempre mon\363tonas (i.e., crescentes ou decrescentes)." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Isso n\343o \351 coi
ncid\352ncia. " }}{PARA 0 "" 0 "" {TEXT -1 254 "Se houvesse uma solu
\347\343o da edo y'=f(y) com derivada nula em algum t* (se a fun\347
\343o n\343o fosse mon\363tona, isso ocorreria - por que?), ter\355amo
s y'(t*)=0 e y'(t*)=f(y(t*)), donde a solu\347\343o constante y(t)=y(t
*) passaria tamb\351m pelo ponto (t*,y(t*)), violando o " }{TEXT 289 
20 "teorema de unicidade" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "Um argumento similar mostra que s
omente solu\347\365es constantes de y'=f(y) atingem seu m\341ximo em t
empo finito. Ou ainda, que nenhuma fun\347\343o peri\363dica \351 solu
\347\343o de y'=f(y). " }}}}{MARK "0 0 0" 19 }{VIEWOPTS 1 1 0 3 2 
1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }