{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
258 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 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 }{CSTYLE "" -1 
263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 
0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
289 "" 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 "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 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT 256 48 "MAT1154 - Equa\347\365es Diferenciais e de Diferen\347as
 " }}{PARA 256 "" 0 "" {TEXT 257 19 "Ass\355ntota vertical." }}{PARA 
256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "" 0 "" {TEXT -1 17 "George S
vetlichny" }}{PARA 256 "" 0 "" {TEXT 258 57 "\005\005\005M\311TODOS NU
M\311RICOS DE SOLU\307\303O DE ODE DE \010PRIMEIRA ORDEM" }}{PARA 256 
"" 0 "" {TEXT 259 69 "Baseado nos scripts do site da Maplesoft e adapt
a\347\365es do Prof. Grivet" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "restart:with( DEtools ):with
( plots ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A e.d.o. " }{TEXT 
261 8 "y'=y^2+t" }{TEXT -1 26 " demonstra o fen\364mento de " }{TEXT 
260 15 "instabilidade. " }{TEXT -1 34 "Definimos primeiro o lado direi
to:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fun:=y(t)^2+t;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Definimos agora a equa\347\343o." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eq:=D(y)(t)=fun;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Achamos a solu\347\343o geral." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sol:=dsolve(eq,y(t));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "A solu\347\343o geral \351 express
a por meio das fun\347\365es especiais de Airy." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 33 "Definimos uma condi\347\343o inical em " }{TEXT 262 
5 "y(0)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ci:=y(0)=0;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Resolvemos agora com esta condi
\347\343o inical:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sola:=
dsolve([eq,ci],y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Pegamos \+
a fun\347\343o solu\347\343o:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "funsol:=subs(sola,y(t));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 26 "Plotamos agora a solu\347\343o  " }{TEXT 271 5 "exata
" }{TEXT -1 14 " no intervalo " }{TEXT 288 5 "[0,3]" }{TEXT -1 2 ".." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(funsol,t=0..3,y=-10.
.10, colour=blue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Note que h
\341 uma ass\355ntota vertical perto do valor " }{TEXT 289 4 "t=2." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Vamos resolver agora a mesma equ
a\347\343o numericamente usando o m\351todo de Euler. Definimos o proc
edimento de Euler. Nesta defini\347\343o " }{TEXT 278 1 "f" }{TEXT -1 
31 " \351 a fun\347\343o do lado direito de " }{TEXT 279 9 "y'=f(t,y)
" }{TEXT -1 2 ", " }{TEXT 280 8 "(t0,y0) " }{TEXT -1 22 "\351 a condi
\347\343o inicial, " }{TEXT 281 1 "h" }{TEXT -1 25 " \351 o tamanho do
 passo, e " }{TEXT 282 1 "T" }{TEXT -1 5 "  (> " }{TEXT 283 3 "t0)" }
{TEXT -1 65 " \351 o ponto final do intervalo onde procuramos a solu
\347\343o n\372merica." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "E
uler := proc( f, t0, y0, h, T )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "
  local i, L, X ,N; N:=round((T-t0)/h);" }{TEXT -1 1 " " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "  X := evalf( [ t0, y0 ] );" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "  L := X;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  f
or i from 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "    X := X +
 [ h, h*f(op(X)) ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "    L := L, \+
X;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "  return L;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{MPLTEXT 1 0 9 "end proc;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 19 "Definimos a fun\347\343o " }{TEXT 263 6 "y^2+t:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "func:=(t,y)->y^2+t;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Chamamos agora o Euler com a mesma
 condi\347\365es iniciais, usand " }{TEXT 286 5 "h=0,3" }{TEXT -1 17 "
 e ponto final 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "aa:=Eu
ler(func,0,0,0.3,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Plotamos \+
a solu\347\343o exata (em azul) junto com o resultado do met\363do num
\351rico (em vermelho)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "
plot([[aa],funsol],t=0..3,y=-5..20,colour=[red,blue]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 77 "Note que o m\351todo num\351rico simples \+
n\343o soube lidar com a ass\355ntota vertical. " }}}}{MARK "27" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }