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with(DEtools): wit h(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 125 "Nesta se\347\343o o nosso \+ objetivo \351 visualizar o m\351todo de Euler para resolver numericame nte problemas de valor inicial da forma" }}{PARA 265 "" 0 "" {XPPEDIT 18 0 "diff(y(t),t) = f(t,y);" "6#/-%%diffG6$-%\"yG6#%\"tGF*-% \"fG6$F*F(" }{TEXT -1 16 " no intervalo " }{XPPEDIT 18 0 "t = a .. b ;" "6#/%\"tG;%\"aG%\"bG" }{TEXT -1 22 " com condi\347\343o inicial " } {XPPEDIT 18 0 "y(a) = y[0];" "6#/-%\"yG6#%\"aG&F%6#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Ap licamos o m\351todo de Euler a casos onde a solu\347\343o exata \351 c onhecida e ent\343o plotamos a solu\347\343o exata e a aproxima\347 \343o." }}{PARA 0 "" 0 "" {TEXT -1 42 "O seguinte procedimento faz est e trabalho." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 682 "euleranimado := proc( f, a, b, y0, num,exact)\nlocal h,n,t1,t2,w1,w2,p,i,across,u p,diag;\nh:=(b-a)/num;\nn := num * 2; \n\011t2 := a;\011w2 := y0;\011 p[0] := pointplot( [t2,w2]):\n\011for i from 1 to n by 2 do\n\011\011t 1 := t2;\011t2 := evalf(t1 + h); \n\011\011w1 := w2;\011w2 := evalf(w 1 + h*f(t1,w1));\n\011\011across := plot( [[t1,w1], [t2,w1]], t = a..b , color = gold ):\n\011\011up := plot([[t2,w1], [t2,w2]], t = a..b, co lor = green):\n\011\011diag:= plot([[t1,w1], [t2,w2]], t = a..b, thick ness= 2, color = red):\n\011\011p[i]:= plots[display](\{p[i-1],across, up\});\n\011\011p[i+1] := plots[display]( \{p[i] , diag \}); od:\n p[n +1]:= plots[display](\{p[n],plot(exact(t),t=a..b)\});\n\011plots[displ ay](seq( p[i], i = 0..n+1), insequence=true ); end:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Vamos aplicar este procedimento a um exemplo simples, usa ndo 50 passos." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "f := (t,y) -> y-t^2+1 ;\ny := t -> (t+1)^2-0.5*exp(t) ; \neuleranimado( f, 0, 4, 0.5 , 50,y) ; " }}}{PARA 264 "" 1 "" {TEXT -1 0 "" }}{PARA 263 "" 1 " " {TEXT -1 0 "" }}{EXCHG {PARA 5 "" 0 "" {TEXT 259 7 "Exemplo" }{TEXT -1 60 ": resolver a EDO u'+2tu= 4t com u(0)=1 no intervalo [0,1]." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "f := (t,y) -> 4*t -2*t*y ;\ny := t -> 2-exp(-t^2);\neuleranimado( f, 0, 1, 1 , 20,y) ; 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