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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 48 "MAT1154 \+
- Equa\347\365es Diferenciais e de Diferen\347as " }}}{EXCHG {PARA 
257 "" 0 "" {TEXT -1 76 "Equa\347\365es  diferenciais e de diferen\347
as lineares de segunda ordem homog\352neas." }{TEXT 257 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "" 0 "" {TEXT -1 17 "George S
vetlichny" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart: with(
DEtools):with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Escrevem
os a forma geral de uma equa\347\343o de segunda ordem linear homog
\352nea." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eq:=(D@@2)(y)(t
)+a(t)*D(y)(t)+b(t)*y(t)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "N
esta forma geral a equa\347\343o n\343o pode ser resolvida, nem por qu
adraturas (com integrais indicadas mas n\343o avaliadas)." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 97 "Vamos portanto resolver alguns exemplos o
nde uma solu\347\343o simb\363lica \351 poss\355vel., por exemplo para
 " }{TEXT 264 8 "y''+ty=0" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "a(t):=0; b(t):=t;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "fun:=subs(dsolve(eq,y(t)),y(t));" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 46 "A solu\347\343o \351 dado pelos fun\347\365es espe
ciais Airy." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Agora vamos plotar
 as duas solu\347\365es linearmente independentes colocando " }{TEXT 
260 5 "(1,0)" }{TEXT -1 3 " e " }{TEXT 261 6 "(0,1) " }{TEXT -1 115 "p
ara as duas constantes livres e depois outras duas solu\347\365es line
armente independentes dando os mesmos valores aos " }{TEXT 262 4 "y(0)
" }{TEXT -1 3 " e " }{TEXT 263 6 "y'(0)." }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fun1:=subs([_C1=1,_C2=0],fun);fun2:
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51 "plot([fun1,fun2],t=-2..2,y=-2..2,color=[blue,red]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "fun3:=subs(dsolve(\{eq,y(0)=1,D(y)
(0)=0\},y(t)),y(t)); fun4:=subs(dsolve(\{eq,y(0)=0,D(y)(0)=1\},y(t)),y
(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([fun3,fun4],t
=-2..2,y=-2..2,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
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{TEXT 265 2 "t " }{TEXT -1 24 "negativo e decrescendo. " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot([fun3,fun4],t=-4..1,color=[blu
e,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Vamos fazer mais uma \+
com coeficientes n\343o constantes." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "a(t):=1/(t+1); b(t):=1/(t+1)^2;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 32 "fun:=subs(dsolve(eq,y(t)),y(t));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fun1:=subs([_C1=1,_C2=0],fun);fun2:
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51 "plot([fun1,fun2],t=-2..3,y=-2..2,color=[blue,red]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "As duas solu\347\365es acima j\341 satisf
azem " }{TEXT 266 20 "(y(0),y'(0)) = (1,0)" }{TEXT -1 3 " e " }{TEXT 
267 5 "(0,1)" }{TEXT -1 47 ". Notem que as solu\347\365es n\343o sao d
efinidas para " }{TEXT 268 8 "t < -1. " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 97 "Vamos agora resolver equa\347\365es com coeficientes cons
tantes incluindo todos os tipos qualitativoes." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 28 "Duas raizes reais positivas:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "a:=-3; b:=2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "fun:=subs(dsolve(eq,y(t)),y(t));" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 41 "As solu\347\365es que correspondem a cada raiz:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fun1:=subs([_C1=1,_C2=0],f
un);fun2:=subs([_C1=0,_C2=1],fun);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "plot([fun1,fun2],t=-2..2,y=-2..2,color=[blue,red]);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "As solu\347\365es que correspon
dem \340s condi\347\365es iniciais " }{TEXT 272 5 "(1,0)" }{TEXT -1 3 
" e " }{TEXT 273 5 "(0,1)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 105 "fun3:=subs(dsolve(\{eq,y(0)=1,D(y)(0)=0\},y(t)),y(
t)); fun4:=subs(dsolve(\{eq,y(0)=0,D(y)(0)=1\},y(t)),y(t));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([fun3,fun4],t=-2..2,y=-
2..2,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Duas r
aizes reais, uma positiva, outra negativa." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "a:=1; b:=-2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "fun:=subs(dsolve(eq,y(t)),y(t));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "As solu\347\365es
 que correspondem a cada raiz:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
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1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "fun3:=subs(dsolve(
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)(0)=1\},y(t)),y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "pl
ot([fun3,fun4],t=-2..2,y=-2..2,color=[blue,red]);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 32 "Uma raiz real negativa repetida." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "a:=1; b:=1/4;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "fun:=subs(dsolve(eq,y(t)),y(t));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 52 "As solu\347\365es padr\343o que correspondem a \+
raiz repetida:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fun1:=sub
s([_C1=1,_C2=0],fun);fun2:=subs([_C1=0,_C2=1],fun);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "plot([fun1,fun2],t=-10..10,y=-2..2,color=
[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "As solu\347\365es
 que correspondem \340s condi\347\365es iniciais " }{TEXT 276 5 "(1,0)
" }{TEXT -1 3 " e " }{TEXT 277 5 "(0,1)" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "fun3:=subs(dsolve(\{eq,y(0)=1,D(y)
(0)=0\},y(t)),y(t)); fun4:=subs(dsolve(\{eq,y(0)=0,D(y)(0)=1\},y(t)),y
(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot([fun3,fun4],t
=-10..10,y=-2..2,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
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{MPLTEXT 1 0 14 "a:=1; b:=17/4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "fun:=subs(dsolve(eq,y(t)),y(t));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 56 "As solu\347\365es padr\343o que correspondem a uma raiz c
omplexa:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fun1:=subs([_C1
=1,_C2=0],fun);fun2:=subs([_C1=0,_C2=1],fun);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 52 "plot([fun1,fun2],t=-2..10,y=-3..2,color=[blue,re
d]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "As solu\347\365es que cor
respondem \340s condi\347\365es iniciais " }{TEXT 278 5 "(1,0)" }
{TEXT -1 3 " e " }{TEXT 279 5 "(0,1)" }{TEXT -1 1 "." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 105 "fun3:=subs(dsolve(\{eq,y(0)=1,D(y)(0)=0
\},y(t)),y(t)); fun4:=subs(dsolve(\{eq,y(0)=0,D(y)(0)=1\},y(t)),y(t));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot([fun3,fun4],t=-2..
10,y=-3..2,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Agora vamos fazer exemplos p
arecidos para equa\347\365es de diferen\347as, lineares, com coeficien
tes constantes, homog\352neas." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eq:=y(n+
2)+a*y(n+1)+b*y(n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsol
ve(eq,y(n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Esta express\343o
 n\343o faz sentido se " }{TEXT 269 9 "a**2 = 4b" }{TEXT -1 83 " o que
 corresponde ao caso de uma raiz real repetida. O limite da express
\343o quando " }{TEXT 271 15 "b -> (1/4)a**2 " }{TEXT -1 35 "existe e \+
d\341 a solu\347\343o naquele caso." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 28 "Duas raizes reais positivos:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "a:=-27/10:b:=9/5: raizes:=solve(z^2+a*z+b=0,z);fun1:=
rsolve(eq,y(n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "As solu\347
\365es que correspondem a cada raiz:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "func1:=raizes[1]^n;func2:=raizes[2]^n;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 15 "A sequ\352ncia de " }{TEXT 280 2 "0 " }
{TEXT -1 2 "a " }{TEXT 281 2 "10" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 53 "seq1:=[[n,func1]$n=0..10]; seq2:=[[n,func2]$n
=0..10];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Plotando:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot([seq1,seq2],x=0..10,style=poin
t,symbol=circle,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 81 "Para visulizar melhor a sequ\352ncia vamos conectar os pontos p
or segmentos de reta." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "pl
ot([seq1,seq2],x=0..10,style=line,color=[blue,red]);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 53 "As sequ\352ncias que correspondem \340s condi
\347\365es iniciais " }{TEXT 287 5 "(1,0)" }{TEXT -1 3 " e " }{TEXT 
288 5 "(0,1)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 97 "seq3:=[[n,subs([y(0)=1,y(1)=0], fun1)]$n=0..10]; seq4:=[[n,subs(
[y(0)=0,y(1)=1], fun1)]$n=0..10];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "plot([seq3,seq4],x=0..10,style=line,color=[blue,red])
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Duas raizes positivas, uma m
aior que " }{TEXT 282 1 "1" }{TEXT -1 14 ", outra menor:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "a:=-2:b:=99/100: raizes:=solve(z^2+
a*z+b=0,z);fun1:=rsolve(eq,y(n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
41 "As solu\347\365es que correspondem a cada raiz:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "func1:=raizes[1]^n;func2:=raizes[2]^n;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "seq1:=[[n,func1]$n=0..10];
 seq2:=[[n,func2]$n=0..10];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "plot([seq1,seq2],x=0..10,style=line,color=[blue,red]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 53 "As sequ\352ncias que correspondem \340s c
ondi\347\365es iniciais " }{TEXT 289 5 "(1,0)" }{TEXT -1 3 " e " }
{TEXT 290 5 "(0,1)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 97 "seq3:=[[n,subs([y(0)=1,y(1)=0], fun1)]$n=0..10]; seq4
:=[[n,subs([y(0)=0,y(1)=1], fun1)]$n=0..10];" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 54 "plot([seq3,seq4],x=0..10,style=line,color=[blue,
red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Uma raiz repetida maior
 que " }{TEXT 283 1 "1" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "a:=-12/5:b:=36/25: raizes:=solve(z^2+a*z+b=0,z);fun1:
=rsolve(eq,y(n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "As solu\347
\365es padr\343o correspondentes a uma raiz repetida." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "func1:=raizes[1]^n;func2:=n*raizes[
2]^n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "seq1:=[[n,func1]$n
=0..10]; seq2:=[[n,func2]$n=0..10];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([s
eq1,seq2],x=0..10,style=line,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 51 "As solu\347\365es que correspondem \340s condi\347
\365es iniciais " }{TEXT 291 4 "(1,0" }{TEXT -1 4 ") e " }{TEXT 292 5 
"(0,1)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "s
eq3:=[[n,subs([y(0)=1,y(1)=0], fun1)]$n=0..10]; seq4:=[[n,subs([y(0)=0
,y(1)=1], fun1)]$n=0..10];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "plot([seq3,seq4],x=0..10,style=line,color=[blue,red]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 28 "Uma raiz repetida menor que " }{TEXT 284 
1 "1" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a:=
-8/5:b:=16/25: raizes:=solve(z^2+a*z+b=0,z);fun1:=rsolve(eq,y(n));" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Solu\347\365es padr\343o correspo
ndentes a uma raiz repetida:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "func1:=raizes[1]^n;func2:=n*raizes[2]^n;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 53 "seq1:=[[n,func1]$n=0..10]; seq2:=[[n,func2]$n=
0..10];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([seq1,seq2],x=0..10,style=line
,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "As solu
\347\365es que correspondem \340s condi\347\365es iniciais " }{TEXT 
294 5 "(1,0)" }{TEXT -1 3 " e " }{TEXT 295 5 "(0,1)" }{TEXT -1 1 "." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "seq3:=[[n,subs([y(0)=1,y(1
)=0], fun1)]$n=0..10]; seq4:=[[n,subs([y(0)=0,y(1)=1], fun1)]$n=0..10]
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([seq3,seq4],x=0..
10,style=line,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
37 "Raizes reais uma positiva, maior que " }{TEXT 285 1 "1" }{TEXT -1 
38 ", outra negativa, de m\363dulo menor que " }{TEXT 286 1 "1" }
{TEXT -1 2 "::" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "a:=-3/5:b
:=-11/20: raizes:=solve(z^2+a*z+b=0,z);fun1:=rsolve(eq,y(n));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "As solu\347\365es que correspondem
 a cada raiz:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "func1:=rai
zes[1]^n;func2:=raizes[2]^n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "seq1:=[[n,func1]$n=0..10]; seq2:=[[n,func2]$n=0..10];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 54 "plot([seq1,seq2],x=0..10,style=line,color=[blue,red
]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "As sequ\352ncias que corre
spondem \340s condi\347\365es iniciais " }{TEXT 296 5 "(1,0)" }{TEXT 
-1 3 " e " }{TEXT 297 5 "(0,1)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 97 "seq3:=[[n,subs([y(0)=1,y(1)=0], fun1)]$n=0..10
]; seq4:=[[n,subs([y(0)=0,y(1)=1], fun1)]$n=0..10];" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 54 "plot([seq3,seq4],x=0..10,style=line,color
=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Raizes complexas
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/36: raizes:=solve(z^2+a*z+b=0,z);fun1:=rsolve(eq,y(n));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 51 "Solu\347\365es padr\343o correspondentes \+
a raizes complexas:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "fun
c1:=(abs(raizes[1])^n)*sin(n*argument(raizes[1])); func2:=(abs(raizes[
1])^n)*cos(n*argument(raizes[1]));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "seq1:=[[n,func1]$n=0..10]; seq2:=[[n,func2]$n=0..10];
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "plot([seq1,seq2],x=0..10,style=line,color=[blu
e,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "As solu\347\365es que
 correspondem \340s condi\347\365es iniciais " }{TEXT 298 5 "(1,0)" }
{TEXT -1 3 " e " }{TEXT 299 5 "(0,1)" }{TEXT -1 1 "." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 97 "seq3:=[[n,subs([y(0)=1,y(1)=0], fun1)]$n=
0..10]; seq4:=[[n,subs([y(0)=0,y(1)=1], fun1)]$n=0..10];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([seq3,seq4],x=0..10,style=line
,color=[blue,red]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{MARK "130" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }