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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 " }}{PARA 257 "" 0 "
" {TEXT -1 65 "Equa\347\365es  diferenciais e de diferen\347as lineare
s de segunda ordem." }{TEXT 257 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "
Oscila\347\365es." }}{PARA 256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "
" 0 "" {TEXT -1 17 "George Svetlichny" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "restart: with(DEtools):with(plots):" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 461 "Um fen\364meno novo que aparece com equa\347
\365es de segunda ordem, em compar\347\343o com as de primeira, \351 a
 presen\347a de solu\347\365es oscilat\363rias de equa\347\365es linea
res homog\352neas com coeficientes constantes. O mesmo sistema real \+
\351 capaz de mostrar v\341rios comportamento qualitativo dependendo d
e valores de par\342metros e condi\347\365es ambientais. Vamos ilustra
r isto com uma equa\347\343o que modela uma massa amarrada a uma mola \+
e que desliza com atrito sobre uma superf\355cie. A constante " }
{TEXT 294 1 "a" }{TEXT -1 1 " " }{TEXT 295 4 "> 0 " }{TEXT -1 29 "dete
rmina a for\347a do atrito.." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "eq:=(D@@2)(y)(t)+0.1*a*D(y)(t)+0.04*y(t)=0;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(t));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 12 "Note que se " }{TEXT 260 5 "a < 4" }{TEXT -1 1 " " }
{TEXT 261 1 " " }{TEXT -1 28 "a raiz quadrada \351 um numero " }{TEXT 
262 0 "" }{TEXT 263 10 "imagin\341rio" }{TEXT 264 2 ". " }{TEXT 266 0 
"" }{TEXT -1 8 "O valor " }{TEXT 267 5 "a = 4" }{TEXT -1 24 " \351 con
hecido como valor " }{TEXT 268 9 "cr\355tico. " }{TEXT -1 0 "" }{TEXT 
265 0 "" }{TEXT -1 47 "Vamos agora plotar as solu\347\365es para valor
es de " }{TEXT 269 2 "a " }{TEXT -1 92 "dos dois lados do valor cr\355
tico e para o valor cr\355tico. Primeiro soltando a mass da posi\347
\343o " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 8 "y(0) = 5" }
{TEXT -1 21 " com velocidade nula " }{TEXT 271 10 "y'(0) = 0." }{TEXT 
-1 77 " Subcr'itico \351 plotado em azul, cr\355tico em vermelho, e su
percr\355tico em verde." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 
"a:=0.5:fun1:=subs(dsolve(\{eq,y(0)=5,D(y)(0)=0\},y(t)),y(t)); a:=4.0:
fun2:=subs(dsolve(\{eq,y(0)=5,D(y)(0)=0\},y(t)),y(t)); a:=7.5:fun3:=su
bs(dsolve(\{eq,y(0)=5,D(y)(0)=0\},y(t)),y(t));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 64 "plot([fun1,fun2,fun3],t=0..100,y=-5..5,color=[
blue, red,green]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Agora vamos
 soltar a massa de novo da posi\347\343o " }{TEXT 272 1 "5" }{TEXT -1 
45 " mas agora com velocidade negativa y'(0) = -3" }{TEXT 273 1 "." }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "a:=0.5:fun
1:=subs(dsolve(\{eq,y(0)=5,D(y)(0)=-3\},y(t)),y(t)); a:=4.0:fun2:=subs
(dsolve(\{eq,y(0)=5,D(y)(0)=-3\},y(t)),y(t)); a:=7.5:fun3:=subs(dsolve
(\{eq,y(0)=5,D(y)(0)=-3\},y(t)),y(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "plot([fun1,fun2,fun3],t=0..100,y=-15..15,color=[blue,
 red,green]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Agora com veloci
dade positiva y'(0) = 3" }{TEXT 274 1 "." }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "a:=0.5:fun1:=subs(dsolve(\{eq,y(0)
=5,D(y)(0)=3\},y(t)),y(t)); a:=4.0:fun2:=subs(dsolve(\{eq,y(0)=5,D(y)(
0)=3\},y(t)),y(t)); a:=7.5:fun3:=subs(dsolve(\{eq,y(0)=5,D(y)(0)=3\},y
(t)),y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([fun1,f
un2,fun3],t=0..100,y=-15..15,color=[blue, red,green]);" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Agora de posi\347\343o " }
{TEXT 276 8 "y(0) = 0" }{TEXT -1 37 ", e com velocidade positiva y'(0)
 = 3" }{TEXT 275 1 "." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 179 "a:=0.5:fun1:=subs(dsolve(\{eq,y(0)=0,D(y)(0)=3\},y(t
)),y(t)); a:=4.0:fun2:=subs(dsolve(\{eq,y(0)=0,D(y)(0)=3\},y(t)),y(t))
; a:=7.5:fun3:=subs(dsolve(\{eq,y(0)=0,D(y)(0)=3\},y(t)),y(t));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([fun1,fun2,fun3],t=0..1
00,y=-15..15,color=[blue, red,green]);" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 66 "Agora vamos ilustrar o mesmo fen\364meno \+
com equa\347\365es de diferen\347as: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "unassign('a'): eq:=y(n+2)-0.15*a*y(n+1)+.09*y(n)=0;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(eq,y(n));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Como antes o valor " }{TEXT 277 6 
"a = 4 " }{TEXT -1 109 "\351 o valor cr\355tico. Abaixo as raizes da e
qua\347\343o caracter\355stica s\343o complexas, acima s\343o reais di
stintas. Para " }{TEXT 278 6 "a = 4 " }{TEXT -1 81 "a express\343o aci
ma n\343o faz sentido por indicar uma divis\343o por zero, mas o limit
e " }{TEXT 279 6 "a -> 4" }{TEXT -1 45 " existe e d\341 a solu\347\343
o para o valor cr\355tico. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Va
mos agora resolver e plotar as solu\347\365es para v\341rios valores d
e " }{TEXT 280 1 "a" }{TEXT -1 45 ", evitando porem raizes com m\363du
lo maior que " }{TEXT 281 1 "1" }{TEXT -1 39 ", ou seja evitando solu
\347\365es crescentes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "
a:=-3.0:fun1:=rsolve(\{eq,y(0)=1,y(1)=1\},y(n)); a:=4.0:fun2:=rsolve(
\{eq,y(0)=1,y(1)=1\},y(n)); a:=6.0:fun3:=rsolve(\{eq,y(0)=1,y(1)=1\},y
(n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "A primeira resposta \351
 escrita como fun\347\343o complexa (pela presen\347a de " }{TEXT 282 
1 "I" }{TEXT -1 40 " que o maple usa para indicar a raiz de " }{TEXT 
283 5 "-1), " }{TEXT -1 125 "mas na realidade a solu\347\343o \351 rea
l pois os dados iniciais s\343o reais. O proximo comando calcula os va
lores de cada fun\347\343o para " }{TEXT 284 1 "n" }{TEXT -1 13 " vari
ando de " }{TEXT 285 1 "0" }{TEXT -1 3 " a " }{TEXT 286 3 "10." }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "seq1:=[[n,f
un1] $n=0..10]:seq2:=[[n,fun2] $n=0..10]:seq3:=[[n,fun3] $n=0..10]:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot([seq1,seq2,seq3],x=0.
.10,style=point, symbol=circle, color=[blue,red,green]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 88 "Para uma visualiza\347\343o melhor vamos \+
replotar e conectar os pontos com segmentos de linha:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot([seq1,seq2,seq3],x=0..10,style
=line, symbol=circle, color=[blue,red,green]);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 59 "Plotamos agora as mesmas solu\347\365es com as condi
\347\365es inciais " }{TEXT 287 6 "y(0)=1" }{TEXT -1 3 " e " }{TEXT 
288 8 "y(1)=0, " }{TEXT -1 15 " e depois com  " }{TEXT 289 6 "y(0)=0" 
}{TEXT -1 3 " e " }{TEXT 290 7 "y(1)=1." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 138 "a:=-3.0:fun1:=rsolve(\{eq,y(0)=1,y(1)=0\},y(n)); a:=
4.0:fun2:=rsolve(\{eq,y(0)=1,y(1)=0\},y(n)); a:=6.0:fun3:=rsolve(\{eq,
y(0)=1,y(1)=0\},y(n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "s
eq1:=[[n,fun1] $n=0..20]:seq2:=[[n,fun2] $n=0..20]:seq3:=[[n,fun3] $n=
0..20]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot([seq1,seq2,seq3],x=
0..10,style=line, symbol=circle, color=[blue,red,green]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "a:=-3.0:fun1:=rsolve(\{eq,y(0)=0,y
(1)=1\},y(n)); a:=4.0:fun2:=rsolve(\{eq,y(0)=0,y(1)=1\},y(n)); a:=6.0:
fun3:=rsolve(\{eq,y(0)=0,y(1)=1\},y(n));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "seq1:=[[n,fun1] $n=0..20]:seq2:=[[n,fun2] $n=0..20]:s
eq3:=[[n,fun3] $n=0..20]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot([
seq1,seq2,seq3],x=0..10,style=line, symbol=circle, color=[blue,red,gre
en]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 313 "Diferentemente das equa
\347\365es diferenciais, uma equa\347\343o de diferen\347as com coefic
ientes constantes poder\341 ter uma solu\347\343o oscilat\363ria mesmo
 sem ter raizes complexas, basta uma raiz ser negativo, ja que pot\352
ncias sucessivas de um n\372mero negativo oscilam de sinal.. Portanto \+
isto j\341 pode acontecer para uma equa\347\343o de " }{TEXT 291 8 "pr
imeira" }{TEXT -1 20 " ordem. Por exemplo." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "eq2:= y(n+1)+0.8*y(n)=0;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 33 "func1:=rsolve(\{eq2,y(0)=1\},y(n));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq1:=[[n,fun1] $n=0..10]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(seq1,x=0..10);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Vejamos o mesmo efeito para uma eq
ua\347\343o de segunda ordem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "eq3:=y(n+2)+0.25*y(n+1)-0.75*y(n)=0;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "func4:=rsolve(\{eq3,y(0)=1,y(1)=0\},y(n));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sequ1:=[[n,func4] $n=0..10];
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(sequ1,x=0..10);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "func5:=rsolve(\{eq3,y(0)=0
,y(1)=1\},y(n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sequ2:=
[[n,func5] $n=0..10];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "pl
ot(sequ2,x=0..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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