{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Cour
ier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu
t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" 
-1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 
2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Para come\347ar, vamos jus
tificar algo que fizemos ao resolver a equa\347\343o log\355stica" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(
y(t),t)=y(t)*(1-y(t))" "6#/-%%diffG6$-%\"yG6#%\"tGF**&-F(6#F*\"\"\",&F
.F.-F(6#F*!\"\"F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 101 "Qua
ndo resolvemos a equa\347\343o acima, simplesmente jogamos fora os m
\363dulos que apareciam ao integrarmos " }{XPPEDIT 18 0 "1/u" "6#*&\"
\"\"F$%\"uG!\"\"" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "1/(1-u)" "6#*&\"\"
\"F$,&F$F$%\"uG!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 
"A solu\347\343o que encontr\341vamos era (" }{TEXT 256 3 "com" }
{TEXT -1 22 " as barras de m\363dulo):" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "abs(y(t)/(1-y(t)))=e^(c0)*e^t" "6
#/-%$absG6#*&-%\"yG6#%\"tG\"\"\",&F,F,-F)6#F+!\"\"F0*&)%\"eG%#c0GF,)F3
F+F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 107 "que a princ\355pio t\355nhamos que analisar perto de y
=1 e y=0 (onde a express\343o dentro do m\363dulo muda de sinal)." }}
{PARA 0 "" 0 "" {TEXT -1 9 "Usando o " }{TEXT 257 33 "teorema de exist
\352ncia e unicidade" }{TEXT -1 74 ", vemos entretanto que a express
\343o nunca muda de sinal. Se para algum t0, " }{XPPEDIT 18 0 "y(t0)/(
1-y(t0))" "6#*&-%\"yG6#%#t0G\"\"\",&F(F(-F%6#F'!\"\"F," }{TEXT -1 68 "
 \351 positivo, para todo t ser\341 positivo (idem para negativo ou ze
ro)." }}{PARA 0 "" 0 "" {TEXT -1 52 "Isso permite que simplesmente tro
quemos a constante " }{TEXT 258 8 "positiva" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "e^(c0)" "6#)%\"eG%#c0G" }{TEXT -1 40 " por uma constant
e com sinal irrestrito " }{XPPEDIT 18 0 "K0" "6#%#K0G" }{TEXT -1 1 ".
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "# conferindo...\nedo:=diff(y(t),t)=
y(t)*(1-y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$edoG/-%%diffG6$-%
\"yG6#%\"tGF,*&F)\"\"\",&F.F.F)!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "dsolve(\{edo,y(0)=y0\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"tG,$*&%#y0G\"\"\",(F*!\"\"-%$expG6#,$F'F-F-
*&F.F+F*F+F+F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Note que a so
lu\347\343o n\343o est\341 definida se o denominador se anular, o que \+
ocorre para t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve(-y0-
exp(-t)+exp(-t)*y0=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#
*&%#y0G\"\"\",&F)!\"\"F(F)F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
208 "Note que somente no trecho 0<y<1 a solu\347\343o estar\341 defini
da para todo tempo (o tempo problem\341tico teria que ser ln de um n
\372mero negativo). Tanto para y0>1 quanto para y0<0 teremos solu\347
\365es ilimitadas perto de " }{XPPEDIT 18 0 "-ln(y0/(-1+y0));" "6#,$-%
#lnG6#*&%#y0G\"\"\",&F)!\"\"F(F)F+F+" }{TEXT -1 62 ", o que veremos ma
is adiante, ao esbo\347ar gr\341ficos de solu\347\365es." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Vamos come\347ar c
om o desenho de uma solu\347\343o que fica acima de y=1." }}{PARA 0 "
" 0 "" {TEXT -1 39 "Para isso, vamos usar um comando novo, " }{TEXT 
260 7 "DEplot," }{TEXT -1 20 " que fica no pacote " }{TEXT 259 7 "DETo
ols" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "wit
h(DETools); # pacote com alguns comandos de edo" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7[s%+AreSimilarG%)DEnormalG%'DEplotG%)DEplot3dG%/DEplot
_polygonG%(DFactorG%,DFactorLCLMG%,DFactorsolsG%+DchangevarG%6Function
DecompositionG%%GCRDG%'GosperG%)HeunsolsG%.HomomorphismsG%3IsHyperexpo
nentialG%%LCLMG%,MeijerGsolsG%<MultiplicativeDecompositionG%.ODEInvari
antsG%0PDEchangecoordsG%5PolynomialNormalFormG%6RationalCanonicalFormG
%/ReduceHyperexpG%-RiemannPsolsG%(XchangeG%,XcommutatorG%'XgaugeG%+Zei
lbergerG%(abelsolG%(adjointG%+autonomousG%-bernoullisolG%)buildsolG%)b
uildsymG%'canoniG%)caseplotG%*casesplitG%*checkrankG%)chinisolG%,clair
autsolG%/constcoeffsolsG%+convertAlgG%+convertsysG%-dalembertsolG%(dco
effsG%*de2diffopG%+dfieldplotG%+diff_tableG%*diffop2deG%/dperiodic_sol
sG%*dpolyformG%&dsubsG%*eigenringG%6endomorphism_charpolyG%'equinvG%&e
ta_kG%*eulersolsG%)exactsolG%(expsolsG%/exterior_powerG%'firintG%(firt
estG%+formal_solG%(gen_expG%,generate_icG%+genhomosolG%'gensysG%-hamil
ton_eqsG%.hypergeomsolsG%)hyperodeG%+indicialeqG%'infgenG%,initialdata
G%/integrate_solsG%*intfactorG%+invariantsG%,kovacicsolsG%-leftdivisio
nG%'liesolG%)line_intG%*linearsolG%)matrixDEG%/matrix_riccatiG%.maxdim
systemsG%-moser_reduceG%)muchangeG%%multG%'mutestG%/newton_polygonG%)n
ormalG2G%*ode_int_yG%'ode_y1G%+odeadvisorG%'odepdeG%.parametricsolG%.p
articularsolG%.phaseportraitG%)poincareG%)polysolsG%1power_equivalentG
%4rational_equivalentG%(ratsolsG%'redodeG%,reduceOrderG%-reduce_orderG
%.regular_partsG%*regularspG%.remove_RootOfG%/riccati_systemG%+riccati
solG%(rifreadG%(rifsimpG%.rightdivisionG%(rtaylorG%-separablesolG%.sin
gularitiesG%,solve_groupG%-super_reduceG%'symgenG%0symmetric_powerG%2s
ymmetric_productG%(symtestG%)transinvG%*translateG%,untranslateG%)varp
aramG%%zoomG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dsolve(\{ed
o,y(0)=y0\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*&%#y
0G\"\"\",(F*!\"\"-%$expG6#,$F'F-F-*&F.F+F*F+F+F-F-" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 186 "# solu\347\343o ilimitada perto de ln((y0-
1)/y0)\ny0:=1.1;\ncondi := [y(0)=y0];\ngsolup:=DEplot( edo, y(t), t=-3
..5, condi,arrows=none,linecolor=black,view=[-5..5,-5..5],thickness=1,
axes=normal):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"#6!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&condiG7#/-%\"yG6#\"\"!$\"#6!\"\"" }
}{PARA 6 "" 1 "" {TEXT -1 69 "Warning, plot may be incomplete, the fol
lowing errors(s) were issued:" }}{PARA 6 "" 1 "" {TEXT -1 82 "   canno
t evaluate the solution further left of -2.3978982, probably a singula
rity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "with(plots): # quer
o usar display" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 85 "tblowup:=ln((y0-1)/y0);\ngreta:=implicitplot(t
=tblowup,t=-5..5,y=-5..5,linestyle=dot):" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(tblowupG$!+t_*yR#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 41 "display(greta,gsolup,view=[-5..5,-5..5]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 344 332 332 {PLOTDATA 2 "6'-%'CURVESG6%X0%)anythingG6\"6#/%.
source_rtableGX0F'F(6\"[gl'!%\"!!#iq\"X\"#!!!!404980000000000000000000
00000000C0032EE3B78624D9C014000000000000C0032EE3B78624D9C013355AF109BA
60C0032EE3B78624D9C012666666666666C0032EE3B78624D9C0119BC1577020C7C003
2EE3B78624D9C010CCCCCCCCCCCDC0032EE3B78624D9C0100227BDD6872DC0032EE3B7
8624D9C00E666666666666C0032EE3B78624D9C00CD11C4879DB27C0032EE3B78624D9
C00B333333333333C0032EE3B78624D9C0099DE91546A7F4C0032EE3B78624D9C00800
0000000000C0032EE3B78624D9C0066AB5E21374C1C0032EE3B78624D9C004CCCCCCCC
CCCDC0032EE3B78624D9C0033782AEE0418EC0032EE3B78624D9C00199999999999AC0
032EE3B78624D9C000044F7BAD0E5BC0032EE3B78624D9BFFCCCCCCCCCCCCDC0032EE3
B78624D9BFF9A23890F3B64EC0032EE3B78624D9BFF6666666666666C0032EE3B78624
D9BFF33BD22A8D4FE8C0032EE3B78624D9BFF0000000000000C0032EE3B78624D9BFE9
AAD7884DD304C0032EE3B78624D9BFE3333333333333C0032EE3B78624D9BFD9BC1577
020C6EC0032EE3B78624D9BFC999999999999AC0032EE3B78624D9BF613DEEB4396A00
C0032EE3B78624D93FC999999999999AC0032EE3B78624D93FD9771DBC3126C6C0032E
E3B78624D93FE3333333333333C0032EE3B78624D93FE9885BAAE56030C0032EE3B786
24D93FF0000000000000C0032EE3B78624D93FF32A943BD9167EC0032EE3B78624D93F
F6666666666666C0032EE3B78624D93FF990FAA23F7CE4C0032EE3B78624D93FFCCCCC
CCCCCCCDC0032EE3B78624D93FFFF76108A5E34BC0032EE3B78624D940019999999999
9AC0032EE3B78624D940032EE3B78624D9C0032EE3B78624D94004CCCCCCCCCCCDC003
2EE3B78624D940066216EAB9580CC0032EE3B78624D94008000000000000C0032EE3B7
8624D94009954A1DEC8B40C0032EE3B78624D9400B333333333333C0032EE3B78624D9
400CC87D511FBE72C0032EE3B78624D9400E666666666666C0032EE3B78624D9400FFB
B08452F1A6C0032EE3B78624D94010CCCCCCCCCCCDC0032EE3B78624D940119771DBC3
126CC0032EE3B78624D94012666666666666C0032EE3B78624D94013310B755CAC06C0
032EE3B78624D940140000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000[gl'!%\"!!#aq
\"T\"#!!!!C0032EE3B78624D9C014000000000000C0032EE3B78624D9C013355AF109
BA60C0032EE3B78624D9C012666666666666C0032EE3B78624D9C0119BC1577020C7C0
032EE3B78624D9C010CCCCCCCCCCCDC0032EE3B78624D9C0100227BDD6872DC0032EE3
B78624D9C00E666666666666C0032EE3B78624D9C00CD11C4879DB27C0032EE3B78624
D9C00B333333333333C0032EE3B78624D9C0099DE91546A7F4C0032EE3B78624D9C008
000000000000C0032EE3B78624D9C0066AB5E21374C1C0032EE3B78624D9C004CCCCCC
CCCCCDC0032EE3B78624D9C0033782AEE0418EC0032EE3B78624D9C00199999999999A
C0032EE3B78624D9C000044F7BAD0E5BC0032EE3B78624D9BFFCCCCCCCCCCCCDC0032E
E3B78624D9BFF9A23890F3B64EC0032EE3B78624D9BFF6666666666666C0032EE3B786
24D9BFF33BD22A8D4FE8C0032EE3B78624D9BFF0000000000000C0032EE3B78624D9BF
E9AAD7884DD304C0032EE3B78624D9BFE3333333333333C0032EE3B78624D9BFD9BC15
77020C6EC0032EE3B78624D9BFC999999999999AC0032EE3B78624D9BF613DEEB4396A
00C0032EE3B78624D93FC999999999999AC0032EE3B78624D93FD9771DBC3126C6C003
2EE3B78624D93FE3333333333333C0032EE3B78624D93FE9885BAAE56030C0032EE3B7
8624D93FF0000000000000C0032EE3B78624D93FF32A943BD9167EC0032EE3B78624D9
3FF6666666666666C0032EE3B78624D93FF990FAA23F7CE4C0032EE3B78624D93FFCCC
CCCCCCCCCDC0032EE3B78624D93FFFF76108A5E34BC0032EE3B78624D9400199999999
999AC0032EE3B78624D940032EE3B78624D9C0032EE3B78624D94004CCCCCCCCCCCDC0
032EE3B78624D940066216EAB9580CC0032EE3B78624D94008000000000000C0032EE3
B78624D94009954A1DEC8B40C0032EE3B78624D9400B333333333333C0032EE3B78624
D9400CC87D511FBE72C0032EE3B78624D9400E666666666666C0032EE3B78624D9400F
FBB08452F1A6C0032EE3B78624D94010CCCCCCCCCCCDC0032EE3B78624D940119771DB
C3126CC0032EE3B78624D94012666666666666C0032EE3B78624D94013310B755CAC06
C0032EE3B78624D94014000000000000-%'COLOURG6&%$RGBG$\"\"\"\"\"!$F4F4F5-
%*LINESTYLEG6#\"\"#-F$6&7S7$$!\"$F4%*undefinedG7$$!+OLLLG!\"*F@7$$!+pm
mmEFDF@7$$!+-+++DFDF@7$$!+NLLLBFD$\"3w?'QwVo$*f\"!#;7$$!+ommm@FD$\"3qu
+Nt3#R%[!#<7$$!+,+++?FD$\"3kmGappFYIFV7$$!+MLLL=FD$\"3,&)=i!Gx!=BFV7$$
!+nmmm;FD$\"3'))fG$y^&z#>FV7$$!+++++:FD$\"3y**)R1h^vo\"FV7$$!+LLLL8FD$
\"3&*[.98XVE:FV7$$!+mmmm6FD$\"3[X]L$\\'H79FV7$$!+#*********!#5$\"3I*[:
m(eAG8FV7$$!+DLLL$)Fbp$\"3/tE(H!*3XE\"FV7$$!+emmmmFbp$\"3(en#G$\\l^@\"
FV7$$!+\"*******\\Fbp$\"3AX*Hln4j<\"FV7$$!+CLLLLFbp$\"3CHpSn(4`9\"FV7$
$!+dmmm;Fbp$\"3fY^3p$=.7\"FV7$$F3FD$\"3')****))*******4\"FV7$$\"+xmmm;
Fbp$\"3oTZ?@\"oL3\"FV7$$\"+WLLLLFbp$\"3Eq7yQznp5FV7$$\"+6+++]Fbp$\"3Oe
VncnNe5FV7$$\"+ymmmmFbp$\"3Ou8#QKf*[5FV7$$\"+XLLL$)Fbp$\"3pGX6mR8T5FV7
$$\"+,+++5FD$\"3*f5-dg+Y.\"FV7$$\"+ommm6FD$\"3o)Rc=-M\"H5FV7$$\"+NLLL8
FD$\"3csUr_:bC5FV7$$\"+-+++:FD$\"3jZ]5(Q/2-\"FV7$$\"+pmmm;FD$\"33X^hK.
Z<5FV7$$\"+OLLL=FD$\"3u+W2,([Z,\"FV7$$\"+.+++?FD$\"3'yL4DLcC,\"FV7$$\"
+qmmm@FD$\"3[5+1HQ_55FV7$$\"+PLLLBFD$\"3F2XL!*Q*)35FV7$$\"+/+++DFD$\"3
*)\\)y)*==v+\"FV7$$\"+rmmmEFD$\"3e:$)pFmN15FV7$$\"+QLLLGFD$\"3Onx6abP0
5FV7$$\"+0+++IFD$\"33o(yuYYX+\"FV7$$\"+smmmJFD$\"3'QNclzXQ+\"FV7$$\"+R
LLLLFD$\"3n'o=y\\`K+\"FV7$$\"+1+++NFD$\"3Kx!*zmDv-5FV7$$\"+tmmmOFD$\"3
L!HtW)*GB+\"FV7$$\"+SLLLQFD$\"3]J6Qh2(>+\"FV7$$\"+2+++SFD$\"3s#R@\"\\w
m,5FV7$$\"+ummmTFD$\"31'e\\h@69+\"FV7$$\"+TLLLVFD$\"3?B@U?V>,5FV7$$\"+
3+++XFD$\"3-_%f<y55+\"FV7$$\"+vmmmYFD$\"32s,v,a&3+\"FV7$$\"+ULLL[FD$\"
3l5$G>&Rs+5FV7$$\"\"&F4$\"3J!>viv71+\"FV-F/6&F1F5F5F5-%&STYLEG6#%%LINE
G-%*THICKNESSG6#F3-%*AXESSTYLEG6#%'NORMALG-%+AXESLABELSG6%%\"tG%\"yG-%
%FONTG6#%(DEFAULTG-%%VIEWG6$;!\"&Fe[lFa]l" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 32 "Agora uma solu\347\343o abaixo de y=0:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "# solu\347\343o ilimitada p
erto de ln((y0-1)/y0)\ny0:=-.1;\ncondi := [y(0)=y0];\ngsoldown:=DEplot
( edo, y(t), t=-3..5, condi,arrows=none,linecolor=red,view=[-5..5,-5..
5],thickness=1,axes=normal):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G
$!\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&condiG7#/-%\"yG6#\"\"!$
!\"\"F," }}{PARA 6 "" 1 "" {TEXT -1 69 "Warning, plot may be incomplet
e, the following errors(s) were issued:" }}{PARA 6 "" 1 "" {TEXT -1 
82 "   cannot evaluate the solution further right of 2.3978958, probab
ly a singularity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "tblowup
:=ln((y0-1)/y0);\ngreta:=implicitplot(t=tblowup,t=-5..5,y=-5..5,color=
blue,linestyle=dot):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(tblowupG$\"
+t_*yR#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display(gret
a,gsoldown,view=[-5..5,-5..5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 417 296 
296 {PLOTDATA 2 "6'-%'CURVESG6%X0%)anythingG6\"6#/%.source_rtableGX0F'
F(6\"[gl'!%\"!!#iq\"X\"#!!!!4049800000000000000000000000000040032EE3B7
8624D9C01400000000000040032EE3B78624D9C013310B755CAC0640032EE3B78624D9
C01266666666666640032EE3B78624D9C0119771DBC3126C40032EE3B78624D9C010CC
CCCCCCCCCD40032EE3B78624D9C00FFBB08452F1A640032EE3B78624D9C00E66666666
666640032EE3B78624D9C00CC87D511FBE7240032EE3B78624D9C00B33333333333340
032EE3B78624D9C009954A1DEC8B4040032EE3B78624D9C00800000000000040032EE3
B78624D9C0066216EAB9580C40032EE3B78624D9C004CCCCCCCCCCCD40032EE3B78624
D9C0032EE3B78624D940032EE3B78624D9C00199999999999A40032EE3B78624D9BFFF
F76108A5E34B40032EE3B78624D9BFFCCCCCCCCCCCCD40032EE3B78624D9BFF990FAA2
3F7CE440032EE3B78624D9BFF666666666666640032EE3B78624D9BFF32A943BD9167E
40032EE3B78624D9BFF000000000000040032EE3B78624D9BFE9885BAAE5603040032E
E3B78624D9BFE333333333333340032EE3B78624D9BFD9771DBC3126C640032EE3B786
24D9BFC999999999999A40032EE3B78624D93F613DEEB4396A0040032EE3B78624D93F
C999999999999A40032EE3B78624D93FD9BC1577020C6E40032EE3B78624D93FE33333
3333333340032EE3B78624D93FE9AAD7884DD30440032EE3B78624D93FF00000000000
0040032EE3B78624D93FF33BD22A8D4FE840032EE3B78624D93FF66666666666664003
2EE3B78624D93FF9A23890F3B64E40032EE3B78624D93FFCCCCCCCCCCCCD40032EE3B7
8624D94000044F7BAD0E5B40032EE3B78624D9400199999999999A40032EE3B78624D9
40033782AEE0418E40032EE3B78624D94004CCCCCCCCCCCD40032EE3B78624D940066A
B5E21374C140032EE3B78624D9400800000000000040032EE3B78624D940099DE91546
A7F440032EE3B78624D9400B33333333333340032EE3B78624D9400CD11C4879DB2740
032EE3B78624D9400E66666666666640032EE3B78624D940100227BDD6872D40032EE3
B78624D94010CCCCCCCCCCCD40032EE3B78624D940119BC1577020C740032EE3B78624
D9401266666666666640032EE3B78624D94013355AF109BA6040032EE3B78624D94014
0000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000[gl'!%\"!!#aq\"T\"#!!!!40032EE3B
78624D9C01400000000000040032EE3B78624D9C013310B755CAC0640032EE3B78624D
9C01266666666666640032EE3B78624D9C0119771DBC3126C40032EE3B78624D9C010C
CCCCCCCCCCD40032EE3B78624D9C00FFBB08452F1A640032EE3B78624D9C00E6666666
6666640032EE3B78624D9C00CC87D511FBE7240032EE3B78624D9C00B3333333333334
0032EE3B78624D9C009954A1DEC8B4040032EE3B78624D9C00800000000000040032EE
3B78624D9C0066216EAB9580C40032EE3B78624D9C004CCCCCCCCCCCD40032EE3B7862
4D9C0032EE3B78624D940032EE3B78624D9C00199999999999A40032EE3B78624D9BFF
FF76108A5E34B40032EE3B78624D9BFFCCCCCCCCCCCCD40032EE3B78624D9BFF990FAA
23F7CE440032EE3B78624D9BFF666666666666640032EE3B78624D9BFF32A943BD9167
E40032EE3B78624D9BFF000000000000040032EE3B78624D9BFE9885BAAE5603040032
EE3B78624D9BFE333333333333340032EE3B78624D9BFD9771DBC3126C640032EE3B78
624D9BFC999999999999A40032EE3B78624D93F613DEEB4396A0040032EE3B78624D93
FC999999999999A40032EE3B78624D93FD9BC1577020C6E40032EE3B78624D93FE3333
33333333340032EE3B78624D93FE9AAD7884DD30440032EE3B78624D93FF0000000000
00040032EE3B78624D93FF33BD22A8D4FE840032EE3B78624D93FF6666666666666400
32EE3B78624D93FF9A23890F3B64E40032EE3B78624D93FFCCCCCCCCCCCCD40032EE3B
78624D94000044F7BAD0E5B40032EE3B78624D9400199999999999A40032EE3B78624D
940033782AEE0418E40032EE3B78624D94004CCCCCCCCCCCD40032EE3B78624D940066
AB5E21374C140032EE3B78624D9400800000000000040032EE3B78624D940099DE9154
6A7F440032EE3B78624D9400B33333333333340032EE3B78624D9400CD11C4879DB274
0032EE3B78624D9400E66666666666640032EE3B78624D940100227BDD6872D40032EE
3B78624D94010CCCCCCCCCCCD40032EE3B78624D940119BC1577020C740032EE3B7862
4D9401266666666666640032EE3B78624D94013355AF109BA6040032EE3B78624D9401
4000000000000-%'COLOURG6&%$RGBG$\"\"!F3F2$\"*++++\"!\")-%*LINESTYLEG6#
\"\"#-F$6&7S7$$!\"$F3$!33DP!)QhlYX!#?7$$!+OLLLG!\"*$!3;B^IN2nv`FC7$$!+
pmmmEFG$!3C51XEK\"oN'FC7$$!+-+++DFG$!3\\NslU-N=vFC7$$!+NLLLBFG$!3[\\8<
i6-%*))FC7$$!+ommm@FG$!3Ah!R:r,C0\"!#>7$$!+,+++?FG$!3Ec\"3l$QkX7Fhn7$$
!+MLLL=FG$!3a`fums)[Z\"Fhn7$$!+nmmm;FG$!39)Q)G@f/Z<Fhn7$$!+++++:FG$!3=
op\"y*>Xq?Fhn7$$!+LLLL8FG$!3#zfU5@q^X#Fhn7$$!+mmmm6FG$!3MW'Rk*GT8HFhn7
$$!+#*********!#5$!3[6**3SK2gMFhn7$$!+DLLL$)Fjp$!3.&HbnT2M6%Fhn7$$!+em
mmmFjp$!3-hVK\\=%f*[Fhn7$$!+\"*******\\Fjp$!3!)\\4Li')oNeFhn7$$!+CLLLL
Fjp$!3K'fE)3cznpFhn7$$!+dmmm;Fjp$!3&)***[fnGoL)Fhn7$$\"\"\"FG$!3+++5,+
++5!#=7$$\"+xmmm;Fjp$!3T&f%Gx<=.7F[s7$$\"+WLLLLFjp$!3'zZffG*4`9F[s7$$
\"+6+++]Fjp$!3Gm^*\\:(4j<F[s7$$\"+ymmmmFjp$!3%\\Qf!RIl^@F[s7$$\"+XLLL$
)Fjp$!3'*3Am!e&4XEF[s7$$\"+,+++5FG$!3Q!f_MymAG$F[s7$$\"+ommm6FG$!3+S<g
D3(H7%F[s7$$\"+NLLL8FG$!35C7_rENk_F[s7$$\"+-+++:FG$!3gi+d\\.`voF[s7$$
\"+pmmm;FG$!37!pW*Regz#*F[s7$$\"+OLLL=FG$!3T2rAyJ3=8!#<7$$\"+.+++?FG$!
3[R*R:\"4HY?Fcv7$$\"+qmmm@FG$!3I4,:_['R%QFcv7$$\"+PLLLBFG$!3!4'e.S`U*
\\\"!#;7$$\"+/+++DFG%*undefinedG7$$\"+rmmmEFGFgw7$$\"+QLLLGFGFgw7$$\"+
0+++IFGFgw7$$\"+smmmJFGFgw7$$\"+RLLLLFGFgw7$$\"+1+++NFGFgw7$$\"+tmmmOF
GFgw7$$\"+SLLLQFGFgw7$$\"+2+++SFGFgw7$$\"+ummmTFGFgw7$$\"+TLLLVFGFgw7$
$\"+3+++XFGFgw7$$\"+vmmmYFGFgw7$$\"+ULLL[FGFgw7$$\"\"&F3Fgw-F/6&F1F4F2
F2-%&STYLEG6#%%LINEG-%*THICKNESSG6#Fhr-%*AXESSTYLEG6#%'NORMALG-%+AXESL
ABELSG6%%\"tG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;!\"&FdzF^\\l" 1 2 0 
1 10 0 2 9 1 4 2 1.000000 43.000000 41.000000 0 0 "Curve 1" "Curve 2" 
}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Finalmente, vamos desenhar v
\341rias solu\347\365es, algumas abaixo, outras acima e as duas solu
\347\365es constantes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "c
ondi := [y(0)=-2,y(0)=0,y(3)=.5,y(0)=1,y(0)=1.1,y(0)=1.5];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&condiG7(/-%\"yG6#\"\"!!\"#/F'F*/-F(6#\"\"
$$\"\"&!\"\"/F'\"\"\"/F'$\"#6F3/F'$\"#:F3" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 134 "DEplot( edo, y(t), t=-5..5, condi,arrows=none,line
color=[red,green,blue,green,black,black],view=[-5..5,-5..5],thickness=
1,axes=boxed);" }}{PARA 6 "" 1 "" {TEXT -1 69 "Warning, plot may be in
complete, the following errors(s) were issued:" }}{PARA 6 "" 1 "" 
{TEXT -1 82 "   cannot evaluate the solution further right of .4054651
1, probably a singularity" }}{PARA 6 "" 1 "" {TEXT -1 69 "Warning, plo
t may be incomplete, the following errors(s) were issued:" }}{PARA 6 "
" 1 "" {TEXT -1 82 "   cannot evaluate the solution further left of -2
.3978982, probably a singularity" }}{PARA 6 "" 1 "" {TEXT -1 69 "Warni
ng, plot may be incomplete, the following errors(s) were issued:" }}
{PARA 6 "" 1 "" {TEXT -1 82 "   cannot evaluate the solution further l
eft of -1.0986125, probably a singularity" }}{PARA 13 "" 1 "" 
{GLPLOT2D 408 380 380 {PLOTDATA 2 "6--%'CURVESG6&7S7$$!+&*******\\!\"*
$!3^A7=-@@7X!#?7$$!+imm\"z%F*$!3ef%zm.pJc&F-7$$!+HLL$e%F*$!3ol?GzxcgoF
-7$$!+'****\\P%F*$!3InZz)e%4j%)F-7$$!+jmmmTF*$!3uSc7rIQW5!#>7$$!+ILLeR
F*$!3)4X=P@1%*G\"FB7$$!+(*****\\PF*$!3o)QVl:?Gf\"FB7$$!+kmmTNF*$!3ggg+
H*>!p>FB7$$!+JLLLLF*$!3Mq#3zF.iV#FB7$$!+)****\\7$F*$!3K)e=4\"Q^<IFB7$$
!+lmm;HF*$!3U>\")*>B+Eu$FB7$$!+KLL3FF*$!3wW)4gR$y\\YFB7$$!+*******\\#F
*$!3Y/%4\"HE8*y&FB7$$!+mmm\"H#F*$!3#3,t***)QpA(FB7$$!+LLL$3#F*$!3u*3yx
Y*R_!*FB7$$!++++v=F*$!3'p<<.c#zQ6!#=7$$!+nmmm;F*$!3*4f=:3i0W\"Fdp7$$!+
MLLe9F*$!3S2')*yQua$=Fdp7$$!+,++]7F*$!3w8<Cm!))4O#Fdp7$$!+ommT5F*$!3md
ZVSu1wIFdp7$$!+ULLL$)!#5$!3eX1v,i>zSFdp7$$!+4++]iF\\r$!3wFh0+ZD[bFdp7$
$!+wmmmTF\\r$!3c=zBhU,TyFdp7$$!+VLL$3#F\\r$!3_v*z4\"3.!=\"!#<7$$!\"\"F
*$!3S+++%*******>F^s7$$\"+BLL$3#F\\r$!3]gBwwW;*e%F^s7$$\"+cmmmTF\\r%*u
ndefinedG7$$\"+*)****\\iF\\rF\\t7$$\"+ALLL$)F\\rF\\t7$$\"+mmmT5F*F\\t7
$$\"+******\\7F*F\\t7$$\"+KLLe9F*F\\t7$$\"+lmmm;F*F\\t7$$\"+)****\\(=F
*F\\t7$$\"+JLL$3#F*F\\t7$$\"+kmm\"H#F*F\\t7$$\"+(******\\#F*F\\t7$$\"+
ILL3FF*F\\t7$$\"+jmm;HF*F\\t7$$\"+'****\\7$F*F\\t7$$\"+HLLLLF*F\\t7$$
\"+immTNF*F\\t7$$\"+&*****\\PF*F\\t7$$\"+GLLeRF*F\\t7$$\"+hmmmTF*F\\t7
$$\"+%****\\P%F*F\\t7$$\"+FLL$e%F*F\\t7$$\"+gmm\"z%F*F\\t7$$\"+$******
*\\F*F\\t-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!FgxFfx-%&STYLEG6#%%LINE
G-%*THICKNESSG6#\"\"\"-F$6&7S7$F(Ffx7$F/Ffx7$F4Ffx7$F9Ffx7$F>Ffx7$FDFf
x7$FIFfx7$FNFfx7$FSFfx7$FXFfx7$FgnFfx7$F\\oFfx7$FaoFfx7$FfoFfx7$F[pFfx
7$F`pFfx7$FfpFfx7$F[qFfx7$F`qFfx7$FeqFfx7$FjqFfx7$F`rFfx7$FerFfx7$FjrF
fx7$F`sFfx7$FesFfx7$FjsFfx7$F^tFfx7$FatFfx7$FdtFfx7$FgtFfx7$FjtFfx7$F]
uFfx7$F`uFfx7$FcuFfx7$FfuFfx7$FiuFfx7$F\\vFfx7$F_vFfx7$FbvFfx7$FevFfx7
$FhvFfx7$F[wFfx7$F^wFfx7$FawFfx7$FdwFfx7$FgwFfx7$FjwFfx7$F]xFfx-F`x6&F
bxFfxFcxFfxFhxF\\y-F$6&7S7$$!+#*******\\F*$\"3u(*4-r#=LN$!#@7$$!+fmm\"
z%F*$\"3]8s77*R(HTF^]l7$$!+ELL$e%F*$\"3=BtgL^\"e3&F^]l7$$!+$****\\P%F*
$\"3)Gx!zxZ;jiF^]l7$$!+gmmmTF*$\"3di)GUsDFr(F^]l7$$!+FLLeRF*$\"3#G-)\\
%o\"f(\\*F^]l7$$!+%*****\\PF*$\"3q\"G]L)f[p6F-7$$!+hmmTNF*$\"3/qAv5z)*
R9F-7$$!+GLLLLF*$\"3#\\V/C=IHx\"F-7$$!+&****\\7$F*$\"3Ml#>5q&p#=#F-7$$
!+imm;HF*$\"3K<\"4W\\/po#F-7$$!+HLL3FF*$\"3qv&4Xa*>2LF-7$$!+'******\\#
F*$\"3Ul;%H1>,2%F-7$$!+jmm\"H#F*$\"3r#*[,J383]F-7$$!+ILL$3#F*$\"3!o*HN
7L*4;'F-7$$!+(****\\(=F*$\"3#>1YF&GAxvF-7$$!+kmmm;F*$\"3xL9L.`$fJ*F-7$
$!+JLLe9F*$\"3M(f3`i-\\9\"FB7$$!+)*****\\7F*$\"3=vZ$=\"3O19FB7$$!+lmmT
5F*$\"3ku-LDT[E<FB7$$!+<LLL$)F\\r$\"3%fjKpK4z6#FB7$$!+%)****\\iF\\r$\"
3uYNh*)Rt&f#FB7$$!+^mmmTF\\r$\"3'*370\")='y<$FB7$$!+=LL$3#F\\r$\"3;zrY
KBL&)QFB7$$\"#:F\\r$\"3^d)f$3#)eUZFB7$$\"+[LL$3#F\\r$\"3)>QR'*z<wx&FB7
$$\"+\"omm;%F\\r$\"3')H&>GH#)=-(FB7$$\"+9++]iF\\r$\"3Cc$=>-4*4&)FB7$$
\"+ZLLL$)F\\r$\"3;UPii3%y-\"Fdp7$$\"+ommT5F*$\"3paxu4iZO7Fdp7$$\"+,++]
7F*$\"3M3=>rLZ![\"Fdp7$$\"+MLLe9F*$\"3eta?@J$Hw\"Fdp7$$\"+nmmm;F*$\"3m
Zwotr3'3#Fdp7$$\"++++v=F*$\"3'f*GcXE&3X#Fdp7$$\"+LLL$3#F*$\"3_JDH;UPcG
Fdp7$$\"+mmm\"H#F*$\"31xY0dMn*H$Fdp7$$\"+*******\\#F*$\"3EVt&\\K0ax$Fd
p7$$\"+KLL3FF*$\"31Tjm[Y%fF%Fdp7$$\"+lmm;HF*$\"3c*fD*Rry\"z%Fdp7$$\"+)
****\\7$F*$\"3<_QLcP47`Fdp7$$\"+JLLLLF*$\"3_h8&=/<d#eFdp7$$\"+kmmTNF*$
\"3t*QN_\\**>K'Fdp7$$\"+(*****\\PF*$\"3=>*o'zly\"z'Fdp7$$\"+ILLeRF*$\"
30<A=jA)yA(Fdp7$$\"+jmmmTF*$\"3\\_O'\\*oTDwFdp7$$\"+'****\\P%F*$\"3avF
kag'=)zFdp7$$\"+HLL$e%F*$\"3Wxr<8)enH)Fdp7$$\"+imm\"z%F*$\"37%*p%4)[Jr
&)Fdp7$$\"+&*******\\F*$\"3eTUM20(z!))Fdp-F`x6&FbxFfxFfxFcxFhxF\\y-F$6
&7S7$F($\"2))***************F^s7$F/$\"2y***************F^s7$F4Fh\\m7$F
9Fh\\m7$F>Fe\\m7$FD$F_yFgx7$FI$\"3A+++++++5F^s7$FN$\"3W+++++++5F^s7$FS
F`]m7$FXF`]m7$FgnF`]m7$F\\oF`]m7$FaoF`]m7$FfoFc]m7$F[pFe\\m7$F`pFh\\m7
$FfpF`]m7$F[qF^]m7$F`qF^]m7$FeqF^]m7$FjqFe\\m7$F`rFh\\m7$Fer$\"2m*****
**********F^s7$FjrF^]m7$F`sF^]m7$FesF^]m7$FjsFd^m7$F^tF^]m7$FatF`]m7$F
dtF^]m7$FgtF`]m7$FjtF`]m7$F]uF^]m7$F`uFc]m7$FcuF`]m7$FfuFc]m7$FiuFc]m7
$F\\vF`]m7$F_vFc]m7$FbvFc]m7$FevF^]m7$FhvF^]m7$F[wF`]m7$F^wF^]m7$FawFe
\\m7$FdwF^]m7$FgwF^]m7$FjwFd^m7$F]xFe\\mFd\\lFhxF\\y-F$6&7S7$F(F\\t7$F
/F\\t7$F4F\\t7$F9F\\t7$F>F\\t7$FDF\\t7$FIF\\t7$FNF\\t7$FSF\\t7$FXF\\t7
$FgnF\\t7$F\\oF\\t7$FaoF\\t7$Ffo$\"3?@$R%=iCA**F^s7$F[p$\"3]Q,Tib;0PF^
s7$F`p$\"3e#))eM4&zbCF^s7$Ffp$\"3'))fG$y^&z#>F^s7$F[q$\"3NFM&pt#\\T;F^
s7$F`q$\"3G0vYQ(yZY\"F^s7$Feq$\"3'pI.s&p.Z8F^s7$Fjq$\"3w&GTN!*3XE\"F^s
7$F`r$\"3%\\&RPMve/7F^s7$Fer$\"3UxLi!=e*f6F^s7$Fjr$\"3Zl$[d'G3E6F^s7$F
`s$\"3')***4,+++5\"F^s7$Fes$\"39'=cSn%pz5F^s7$Fjs$\"3abg=H:vj5F^s7$F^t
$\"3]ty/))*[60\"F^s7$Fat$\"3cEI@mR8T5F^s7$Fdt$\"3GTdY+<9L5F^s7$Fgt$\"3
#R!QMABuE5F^s7$Fjt$\"3[0K*pI/;-\"F^s7$F]u$\"3(3D'oK.Z<5F^s7$F`u$\"3$*
\\B'QIQT,\"F^s7$Fcu$\"3tAx=6*[9,\"F^s7$Ffu$\"3^x>-&zv#45F^s7$Fiu$\"3ey
=$**==v+\"F^s7$F\\v$\"3+s^6Qc415F^s7$F_v$\"3s]g5'eV\\+\"F^s7$Fbv$\"3oP
\"yR25S+\"F^s7$Fev$\"3yG8&y\\`K+\"F^s7$Fhv$\"34*)4K=*RE+\"F^s7$F[w$\"3
=SPT$QU@+\"F^s7$F^w$\"30%4<1vQ<+\"F^s7$Faw$\"3Ofz;;7T,5F^s7$Fdw$\"3*zD
K&Hb9,5F^s7$Fgw$\"3\\&>w7&)H4+\"F^s7$Fjw$\"3-\"p*[!ya2+\"F^s7$F]x$\"3a
\"[ziv71+\"F^s-F`x6&FbxFfxFfxFfxFhxF\\y-F$6&7SFc`mFd`mFe`mFf`mFg`mFh`m
Fi`mFj`mF[amF\\amF]amF^amF_am7$FfoF\\t7$F[pF\\t7$F`pF\\t7$FfpF\\t7$F[q
F\\t7$F`qF\\t7$Feq$\"3M/6;m.`1=!#;7$Fjq$\"3[I4S.'*p\"H%F^s7$F`r$\"3sO3
*e,^2l#F^s7$Fer$\"3wgwA4uyA?F^s7$Fjr$\"37s&R38rkp\"F^s7$F`s$\"31++v+++
+:F^s7$Fes$\"3mDWeG_2r8F^s7$Fjs$\"3/EG)>KN;G\"F^s7$F^t$\"3!4#o3Nu;<7F^
s7$Fat$\"3YW$Q$ysSp6F^s7$Fdt$\"3U*yvX+,L8\"F^s7$Fgt$\"35@J\\y\\e06F^s7
$Fjt$\"3-V7Ad\"fS3\"F^s7$F]u$\"3ERJ%Qd)=n5F^s7$F`u$\"3[2_Wm?(Q0\"F^s7$
Fcu$\"3GAl7z=IV5F^s7$Ffu$\"35aHF*Ru[.\"F^s7$Fiu$\"3O\">B'\\78G5F^s7$F
\\v$\"3N]U^#\\?F-\"F^s7$F_v$\"3NNe@W\"p$=5F^s7$Fbv$\"3^;GW=J'[,\"F^s7$
Fev$\"3Z4.+/V.75F^s7$Fhv$\"3VhL*3#*[(45F^s7$F[w$\"3wzw'z/,z+\"F^s7$F^w
$\"35O*o]_0k+\"F^s7$Faw$\"3an>;`Y>05F^s7$Fdw$\"3Mm**p5N@/5F^s7$Fgw$\"3
:,O>,%=M+\"F^s7$Fjw$\"3)\\ECGptF+\"F^s7$F]x$\"3(=N3&H3D-5F^sF\\hmFhxF
\\y-%%VIEWG6$;$!\"&Fgx$\"\"&Fgx;$!+xW;*e%F*$\"+m.`1=FexF\\y-%*AXESSTYL
EG6#%$BOXG-%+AXESLABELSG6$Q\"t6\"Q%y(t)Fg_n-Fc^n6$;Fg^nFi^nF[`n" 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" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Ok, agora m
udando de assunto..." }}{PARA 0 "" 0 "" {TEXT -1 63 "Vimos v\341rios e
xemplos concretos de como resolver EDO's da forma" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(y(x),x)+a(x)*y(x)=
b(x)" "6#/,&-%%diffG6$-%\"yG6#%\"xGF+\"\"\"*&-%\"aG6#F+F,-F)6#F+F,F,-%
\"bG6#F+" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "y(x0)=y0" "6#/-%\"yG6#%#x0
G%#y0G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 82 "Na verdade, gostar\355amos de ter uma f\363rmula par
a a solu\347\343o geral do problema acima." }}{PARA 0 "" 0 "" {TEXT 
-1 78 "A boa not\355cia \351 que o problema \351 t\343o f\341cil, que \+
at\351 o Maple sabe essa f\363rmula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "edo:=diff(y(x),x)+a(x)*y(
x) = b(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$edoG/,&-%%diffG6$-%\"
yG6#%\"xGF-\"\"\"*&-%\"aGF,F.F*F.F.-%\"bGF," }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "dsolve(\{edo,y(x0)=y0\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&-%$IntG6$*&-%\"bG6#%$_z1G\"\"\"-%$exp
G6#-F+6$-%\"aGF0/F1;%#x0GF1F2/F1;F<F'F2#F2\"#5!\"\"F2-F46#-F+6$,$F8FAF
=F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Ok, j\341 temos agora uma
 f\363rmula para edo's dessa forma (que se chamam, como diria o Lula, \+
en passent, " }{TEXT 261 20 "lineares de ordem um" }{TEXT -1 2 ")." }}
{PARA 0 "" 0 "" {TEXT -1 63 "Vamos treinar agora com alguns problemas \+
concretos (num\351ricos)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "a) Uma massa \351
 largada do alto de um edif\355cio e cai sujeita \340 for\347a da grav
idade e do atrito do ar, assumido proporcional ao m\363dulo da velocid
ade." }}{PARA 0 "" 0 "" {TEXT -1 61 "Ache a equa\347\343o de movimento
, sua solu\347\343o e esboce um gr\341fico." }}{PARA 0 "" 0 "" {TEXT 
-1 44 "m=0.25kg, v0=0, k=1/30 (atrito), g=9.8 m/s^2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "m:=.25;\ng:=9.8;\nv0:=0;\nk:=1/30.;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"#D!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG$\"#)*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#v0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"+LLLLL!#6" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "edo1:=m*diff(v(t),t)=m*g-k*v
(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%edo1G/,$*&$\"#D!\"#\"\"\"-%
%diffG6$-%\"vG6#%\"tGF2F+F+,&$\"%]C!\"$F+*&$\"+LLLLL!#6F+F/F+!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dsolve(\{edo1,v(0)=v0\});" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"tG,&#\"-++++]C\"+LLLLL\"
\"\"*&#\"-++++]CF+F,-%$expG6#,$*(\"+LLLLLF,\",++++]#!\"\"F'F,F7F,F7" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sol1:=rhs(%);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%sol1G,&#\"-++++]C\"+LLLLL\"\"\"*&#\"-++++]CF
(F)-%$expG6#,$*(\"+LLLLLF)\",++++]#!\"\"%\"tGF)F4F)F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(sol1,t=0..100);" }}{PARA 13 "
" 1 "" {GLPLOT2D 506 286 286 {PLOTDATA 2 "6%-%'CURVESG6$7Z7$$\"\"!F)F(
7$$\"3)pmm;a)G\\a!#=$\"3erO_7;\"4:&!#<7$$\"3SLLL3x&)*3\"F0$\"3.I&eQ7X3
%**F07$$\"3******\\ilyM;F0$\"3)34fM!)4&R9!#;7$$\"3ymmm;arz@F0$\"3+l&e&
)\\>P&=F;7$$\"3(****\\7y%*z7$F0$\"3Ck(*oxd^1DF;7$$\"3eLL$e9ui2%F0$\"3s
@VA\"3z<3$F;7$$\"3C++voMrU^F0$\"3ehuF:9^ZOF;7$$\"3*omm;z_\"4iF0$\"32'*
>fD'e#QTF;7$$\"3=nmmm6m#G(F0$\"31KG%4N%emXF;7$$\"3OommT&phN)F0$\"3#RN*
e/vyP\\F;7$$\"3x++v=ddC%*F0$\"3=dR<ZM1e_F;7$$\"3KLLe*=)H\\5F;$\"3$Q.P7
\\:e`&F;7$$\"3imm\"z/3uC\"F;$\"3-&zOfOep&fF;7$$\"3/++DJ$RDX\"F;$\"3u^Z
)>#\\I!H'F;7$$\"3wmm\"zR'ok;F;$\"3m53S#>*Q^lF;7$$\"39++D1J:w=F;$\"3--)
HY9-wu'F;7$$\"3oLLL3En$4#F;$\"3KpAZ_$f#**oF;7$$\"3-nm;/RE&G#F;$\"3,vqc
YD(3+(F;7$$\"3_+++D.&4]#F;$\"3))*3=w<G\")3(F;7$$\"32+++vB_<FF;$\"3k2ks
k#3Q:(F;7$$\"3W+++v'Hi#HF;$\"3c_Q?$ol9?(F;7$$\"3%pm;z*ev:JF;$\"3H8P5dM
jMsF;7$$\"3)RLL$347TLF;$\"3q9t\\ZfdksF;7$$\"3KLLLLY.KNF;$\"3c&=%f0Qx$G
(F;7$$\"3O++D\"o7Tv$F;$\"3@=l6:pu+tF;7$$\"3kLLL$Q*o]RF;$\"3Z^PH%3.@J(F
;7$$\"3m++D\"=lj;%F;$\"3JuMJjSd@tF;7$$\"3A++vV&R<P%F;$\"3G;q(yA$QGtF;7
$$\"31ML$e9Ege%F;$\"3r/8,1bvLtF;7$$\"3qLLeR\"3Gy%F;$\"3#o1-@Q/vL(F;7$$
\"35nm;/T1&*\\F;$\"3X3RrzVeStF;7$$\"3=nm\"zRQb@&F;$\"3by`@>D)HM(F;7$$
\"3^++v=>Y2aF;$\"3:1-y>pcWtF;7$$\"3%pmm\"zXu9cF;$\"3[UA7i)yeM(F;7$$\"3
F+++]y))GeF;$\"3'GX//Z-pM(F;7$$\"3I++]i_QQgF;$\"3a78OotlZtF;7$$\"3>++D
\"y%3TiF;$\"3u.?*)e@@[tF;7$$\"3m****\\P![hY'F;$\"3w9\\iYcn[tF;7$$\"3ML
LL$Qx$omF;$\"3EM>1^'))*[tF;7$$\"3%3++]P+V)oF;$\"31(>K<lT#\\tF;7$$\"3Am
m\"zpe*zqF;$\"3cM,c(y:%\\tF;7$$\"3;,++D\\'QH(F;$\"35Qagb2c\\tF;7$$\"3y
KLe9S8&\\(F;$\"3YO;CRTm\\tF;7$$\"3!4+]i?=bq(F;$\"3i!*e2#HY(\\tF;7$$\"3
4LLL3s?6zF;$\"3SF'4l92)\\tF;7$$\"3h++DJXaE\")F;$\"35+!G#y_&)\\tF;7$$\"
3&zmmm'*RRL)F;$\"3]B4JT-*)\\tF;7$$\"3)ommTvJga)F;$\"3o-%)Qxs\"*\\tF;7$
$\"3'[L$e9tOc()F;$\"3K!4Tx]P*\\tF;7$$\"33,++]Qk\\*)F;$\"3kF!eXq^*\\tF;
7$$\"3uLL$3dg6<*F;$\"3s*3\"\\bS'*\\tF;7$$\"3]nmmmxGp$*F;$\"3'e0?/Ss*\\
tF;7$$\"3K,+D\"oK0e*F;$\"3/#*>Yv\"z*\\tF;7$$\"3Y,+v=5s#y*F;$\"3%e+Tl4%
)*\\tF;7$$\"$+\"F)$\"3ge&)p'4))*\\tF;-%'COLOURG6&%$RGBG$\"*++++\"!\")F
(F(-%+AXESLABELSG6$%\"tGQ!6\"-%%VIEWG6$;F(Ff\\l%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(sol1,t=infinity);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6##\"-++++]C\"+LLLLL" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+
,++]t!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Vemos, tanto pela f
\363rmula da solu\347\343o, como pelo gr\341fico, que a velocidade n
\343o cresce indefinidamente, mas tem um limite assint\363tico, chamad
o de " }{TEXT 262 19 "velocidade terminal" }{TEXT -1 25 ", que no caso
 \351 73.5 m/s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 48 "Considere agora v0=-20 m/s (jogamos para cima). " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "dsolve(\{edo1,v(0)=-20\});" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"vG6#%\"tG,&#\"-++++]C\"+LLLLL\"
\"\"*&#\"-gmmm;JF+F,-%$expG6#,$*(\"+LLLLLF,\",++++]#!\"\"F'F,F7F,F7" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sol2:=rhs(%);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%sol2G,&#\"-++++]C\"+LLLLL\"\"\"*&#\"-gmmm;JF
(F)-%$expG6#,$*(\"+LLLLLF)\",++++]#!\"\"%\"tGF)F4F)F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(sol2,t=0..10);" }}{PARA 13 "" 
1 "" {GLPLOT2D 624 312 312 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$!#?
F)7$$\"3emmm;arz@!#=$!3_GQU3H<K<!#;7$$\"3[LL$e9ui2%F/$!3gB&3eT(Q0:F27$
$\"3nmmm\"z_\"4iF/$!3Ek7^\"o,rD\"F27$$\"39ommT&phN)F/$!3,]jZp/?95F27$$
\"3KLLe*=)H\\5!#<$!3II5o'e,Ez(FE7$$\"3smm\"z/3uC\"FE$!3e5#>1P)RtcFE7$$
\"3!****\\7LRDX\"FE$!3kR(e@+&GPNFE7$$\"3%om;zR'ok;FE$!3oj$\\x'yr)Q\"FE
7$$\"33++D1J:w=FE$\"3%)QZ!*z[@LpF/7$$\"3oLLL3En$4#FE$\"3#p1&HwwduFFE7$
$\"3#pmmT!RE&G#FE$\"3a0]'>*GTeXFE7$$\"3D+++D.&4]#FE$\"3ARFqX4\"G^'FE7$
$\"3;+++vB_<FFE$\"3/=6)y%**[>%)FE7$$\"33+++v'Hi#HFE$\"3<W//jgb?5F27$$
\"3&om;z*ev:JFE$\"3!R&\\qF\")\\y6F27$$\"3_LLL347TLFE$\"3%fQ?Q/&=h8F27$
$\"3nLLLLY.KNFE$\"3gTQ_$R2<^\"F27$$\"33++D\"o7Tv$FE$\"3S#f9%[s/#o\"F27
$$\"3?LLL$Q*o]RFE$\"3dC^1,cnG=F27$$\"3m++D\"=lj;%FE$\"3Q>)=b')*=&)>F27
$$\"3S++vV&R<P%FE$\"3#Gd@tk-,8#F27$$\"3CML$e9Ege%FE$\"3C=j')[F8xAF27$$
\"3]LLeR\"3Gy%FE$\"3.P%ywm,&3CF27$$\"3emm;/T1&*\\FE$\"3)\\wPxi*QYDF27$
$\"3=nm\"zRQb@&FE$\"33t3MdRa&o#F27$$\"3:++v=>Y2aFE$\"3C9iE!=#R.GF27$$
\"3Znm;zXu9cFE$\"33Dc`^$Ht#HF27$$\"34+++]y))GeFE$\"3)Q\\!=O;#=0$F27$$
\"3H++]i_QQgFE$\"3kX'RgF@-<$F27$$\"3b++D\"y%3TiFE$\"3S<'\\5it;G$F27$$
\"3+++]P![hY'FE$\"3],:CNT%>S$F27$$\"3iKLL$Qx$omFE$\"3i'4\\.wwp]$F27$$
\"3Y+++v.I%)oFE$\"31<8_-*Qgh$F27$$\"3?mm\"zpe*zqFE$\"3G'4Tt+!>7PF27$$
\"3;,++D\\'QH(FE$\"3$>&[:pxZ9QF27$$\"3%HL$e9S8&\\(FE$\"3.'oc![[43RF27$
$\"3s++D1#=bq(FE$\"3WjmV6HI.SF27$$\"3\"HLL$3s?6zFE$\"3s(R'e\"eRQ4%F27$
$\"3a***\\7`Wl7)FE$\"3mOxvG***f=%F27$$\"3enmmm*RRL)FE$\"3'fc8F^%HsUF27
$$\"3%zmmTvJga)FE$\"3%fm:tb4\"eVF27$$\"3]MLe9tOc()FE$\"3/2S4,2&3W%F27$
$\"31,++]Qk\\*)FE$\"31u&y2Li[^%F27$$\"3![LL3dg6<*FE$\"3spwK(=vtf%F27$$
\"3%ymmmw(Gp$*FE$\"3!QXU)*4R\"pYF27$$\"3C++D\"oK0e*FE$\"3Ya&)=<ZfVZF27
$$\"35,+v=5s#y*FE$\"3%*H'z=p?H\"[F27$$\"#5F)$\"3Y#fI)enO&)[F2-%'COLOUR
G6&%$RGBG$\"*++++\"!\")F(F(-%+AXESLABELSG6$%\"tGQ!6\"-%%VIEWG6$;F(Fez%
(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu
rve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# tempo de altura
 m\341xima(v=0)\ntmax:=fsolve(sol2=0,t=0..2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%tmaxG$\"+E-20=!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "# altura m\341xima relativa ao ponto de largada\nint(
sol2,t=0..tmax);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+UQtK<!\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# velocidade terminal\nlimit
(sol2,t=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"-++++]C\"+LLL
LL" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#$\"+,++]t!\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "41" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }