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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 20 "Equa\347\365es Separ
\341veis\n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Um problema do liv
ro..." }}{PARA 0 "" 0 "" {TEXT -1 76 "Vejamos como fariamos o exemplo \+
1 da secao 2.2 com a ajuda do Maple.\nA EDO \351" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "with(plots):with(DETools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "edo:=diff(y(x),x)=x^2/(1-y(x)^2);" }}{PARA 11 "" 1 "
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o de dire\347\365es:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "cam
po:=DEplot(edo,y(x),x=-3..3,y=-3..3):\ndisplay(campo,scaling=constrain
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jelFgel" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curv
e 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "Note que a equa\347\343o
 diferencial n\343o \351 das mais \"comportadas\": quando o denominado
r 1-y^2 se anula (isto \351, quando y=1 ou y=-1), o campo de dire\347
\365es tende a ficar vertical." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 97 "Como explicado no livro, as curvas de n
\355vel da seguinte fun\347\343o sao solu\347\365es (em forma impl\355
cita):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=-x^3+3*y-y^3;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*$)%\"xG\"\"$\"\"\"!\"\"*&
F)F*%\"yGF*F**$)F-F)F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Para \+
ver as curvas de nivel dessa funcao com o comando abaixo:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "curvas:=contourplot(f, x=-3..3, y=-
3..3, contours=60):\ndisplay(curvas,scaling=constrained);" }}{PARA 13 
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17
" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C
urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve
 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36
" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "C
urve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve
 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55
" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "...e outro n\343o (mas
 com a mesma EDO)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Vamos agora \+
resolver um problema envolvendo essa equa\347\343o que n\343o est\341 \+
no livro:\nRESOLVER o problema de valor inicial " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "diff(y(x),x)=x^2/(1-y
^2)" "6#/-%%diffG6$-%\"yG6#%\"xGF**&F*\"\"#,&\"\"\"F.*$F(F,!\"\"F0" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "y(0)=1.5" "6#/-%\"yG6#\"\"!-%&Float
G6$\"#:!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 215 "E determinar tamb\351m o intervalo m\341ximo d
os x onde est\341 definida uma solu\347\343o y(x).\nSolu\347\343o: a s
olu\347\343o \351 dada em forma impl\355cita pela equa\347\343o f(x,y)
=c, com c escolhido de modo a satisfazer a condi\347\343o inicial. Por
tanto" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "c:=subs([x=0,y=1.5
],f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"%D6!\"$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 84 "Podemos ver o gr\341fico dessa curva (iso
lada das demais curvas que vimos acima) assim:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 96 "solimp:=implicitplot(f=c, x=-3..3, y=-3..3,num
points=5000):\ndisplay(solimp,scaling=constrained);" }}{PARA 13 "" 1 "
" {GLPLOT2D 346 248 248 {PLOTDATA 2 "6%-%'CURVESG6$X0%)anythingG6\"6#/
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#%,CONSTRAINEDG-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 1 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 515 "Por\351m isso n\343o define uma \372nica fun\347\343o y(
x). Temos que cortar a curva e selecionar o peda\347o que passa pelo p
onto (0,1.5). Os pontos de corte s\343o aqueles onde a reta tangente \+
\340 curva fica vertical (lembre do que voce aprendeu sobre fun\347
\365es impl\355citas em C\341lculo 2). Esses pontos s\343o exatamente \+
aqueles onde o denominador na EDO 1-y^2 se anula, isto eh, quando y=1 \+
ou y=-1. Para achar os valores de x, fazemos assim: (s\363 h\341 um va
lor de x para cada caso, portanto n\343o \351 necess\341rio dar o inte
rvalo de busca para fsolve)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "xesquerda:=fsolve(subs(y=-1,f)=c);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%*xesquerdaG$!+p)3?Y\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "xdireita:=fsolve(subs(y=1,f)=c);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)xdireitaG$\"+9flk&*!#5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 196 "Cortando a curva nesses dois pontos, ela fica separada e
m tr\352s peda\347os que s\343o gr\341ficos de fun\347\365es. O peda
\347o que nos interessa \351 o de cima. Mais uma vez, o gr\341fico ago
ra com os pontos encontrados:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 132 "pontos:=pointplot([[xesquerda,-1],[xdireita,1]],symbol=circle,s
ymbolsize=15,color=blue):\ndisplay(solimp,pontos,scaling=constrained);
" }}{PARA 13 "" 1 "" {GLPLOT2D 286 212 212 {PLOTDATA 2 "6&-%'CURVESG6$
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45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
61 "RESPOSTA DO PROBLEMA: A solu\347\343o \351 dada em forma impl\355c
ita por" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 " -x^3+3*y-y^3 = 1.375" "6#
/,(*$%\"xG\"\"$!\"\"*&F'\"\"\"%\"yGF*F**$F+F'F(-%&FloatG6$\"%v8!\"$" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 68 "o intevalo m\341ximo do
s x onde a solu\347\343o est\341 definida \351 o intervalo (" }
{XPPEDIT 18 0 "-infinity,0.956" "6$,$%)infinityG!\"\"-%&FloatG6$\"$c*!
\"$" }{TEXT -1 90 ").\nObs: tomamos um intervalo aberto, pois no extre
mo aparece uma divis\343o por zero na EDO.\n" }}{PARA 0 "" 0 "" {TEXT 
-1 12 "Continua\347\343o:" }}{PARA 0 "" 0 "" {TEXT -1 241 "Quando isol
ar o y em fun\347\343o de x \351 complicado, como nesse caso, nos cont
entamos com a solu\347\343o em foma impl\355cita. Mas podemos tentar v
er se o Maple consegue isolar y. Para isso, temos que usar o comando s
olve (e nao o fsolve, que \351 num\351rico):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "solve(f=c,y);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
%,&*&$\"+++++D!#5\"\"\"),($\"#O\"\"!!\"\"*&$\"#KF-F()%\"xG\"\"$F(F.*&$
\"\"%F-F(,($\"$v\"F-F.*&$\"$W\"F-F(F2F(F(*&$\"#kF-F()F3\"\"'F(F(#F(\"
\"#F(#F(F4F(F(*&F6F(F*#F.F4F(,(*&$\"++++]7F'F(F)F(F.*&$FDF-F(F*FGF.*&^
#$\"+>q7IVF'F(,&*&$\"+++++]F'F(F)F(F(*&$\"\")F-F(F*FGF.F(F(,(*&$\"++++
]7F'F(F)F(F.*&$FDF-F(F*FGF.*&^#$!+>q7IVF'F(FRF(F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 107 "Cuidado: aparecem 3 solu\347\365es, e 2 delas env
olvem o n\372mero imagin\341rio I. Vamos considerar apenas a primeira.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "g:=.2500000000*(-36.-3
2.*x^3+4.*sqrt(-175.+144.*x^3+64.*x^6))^(1/3)+4./(-36.-32.*x^3+4.*sqrt
(-175.+144.*x^3+64.*x^6))^(1/3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
\"gG,&*&$\"+++++D!#5\"\"\"),($\"#O\"\"!!\"\"*&$\"#KF/F*)%\"xG\"\"$F*F0
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\"\"#F*#F*F6F*F**&F8F*F,#F0F6F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 87 "cima:=plot(g,x=-3..xdireita,color=blue,thickness=2):\ndisplay(
cima,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 262 262 262 
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{TEXT -1 107 "Este \351 o \"peda\347o de cima\" da curva vista antes, \+
e \351 o gr\341fico da solu\347\343o y(x) do problema de valor inicial
." }}{PARA 0 "" 0 "" {TEXT -1 54 " Para concluir, vamos sobrepor com o
 gr\341fico anterior:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "di
splay(solimp,cima,pontos,scaling=constrained);" }}{PARA 13 "" 1 "" 
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