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{SECT 0 {EXCHG {PARA 219 "" 0 "" {TEXT 204 48 "MAT1154 - Equa\347\365e
s Diferenciais e de Diferen\347as " }}{PARA 219 "" 0 "" {TEXT 205 47 "
Solu\347\343o de Equa\347\365es Diferenciais do Tipo Linear" }}{PARA 
220 "" 0 "" }{PARA 220 "" 0 "" {TEXT 206 16 "Acknowledgement:" }{TEXT 
207 71 " Este script foi baseado em um semelhante obtido no site da Ma
pleSoft. " }}{PARA 220 "" 0 "" {TEXT 206 12 "Recomenda\347\343o" }
{TEXT 207 93 "      : Ao aluno \351 fortemente recomendado que ele nav
egue no Help do Maple e veja informa\347\365es" }}{PARA 220 "" 0 "" 
{TEXT 207 78 "                                 mais detalhadas sobre o
s comandos utilizados." }}{PARA 220 "" 0 "" {TEXT 207 28 "            \+
                " }}}{SECT 0 {PARA 221 "" 0 "" {TEXT 208 15 "1.Initial
iza\347\343o" }}{PARA 222 "" 0 "" {TEXT 200 117 "Inicialmente devemos \+
remover todos os vestigios de objetos previamente definidos e disponib
ilizar as ferramentas que " }}{PARA 222 "" 0 "" {TEXT 200 24 "usaremos
 durante a se\347\343o" }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 209 8 
"restart:" }}{PARA 223 "> " 0 "" {MPLTEXT 1 209 16 "with( DEtools ):" 
}}{PARA 223 "> " 0 "" {MPLTEXT 1 209 14 "with( plots ):" }}{PARA 223 "
> " 0 "" {MPLTEXT 1 209 15 "with( linalg ):" }}}}{PARA 222 "" 0 "" }
{SECT 0 {PARA 221 "" 0 "4.A" {TEXT 208 47 "2.Estrutura de Solu\347\343
o Geral para EDOs Lineares" }}{EXCHG {PARA 220 "" 0 "" {TEXT 207 67 "U
ma EDO linear de primeira ordem possui a seguinte express\343o geral "
 }{TEXT 210 21 "x\264(t)+p(t).x(t)=g(t)." }}{PARA 220 "" 0 "" }{PARA 
223 "> " 0 "" {MPLTEXT 1 0 48 "lin_ode := diff( x(t), t ) + p(t) * x(t
) = f(t);" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 207 60 "Observe que esta \+
EDO em geral n\343o \351 de vari\341veis separ\341veis:" }}{PARA 220 "
" 0 "" }{PARA 223 "> " 0 "" {MPLTEXT 1 0 35 "odeadvisor( lin_ode, [sep
arable] );" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 207 14 "Entretando se " 
}{XPPEDIT 18 0 "p(t)" "6#-%\"pG6#%\"tG" }{TEXT 207 3 " e " }{XPPEDIT 
18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT 207 86 " s\343o constantes, a ODE
 \351 de vari\341veis separ\341veis e sendo assim ela pode ser resolvi
da " }}{PARA 220 "" 0 "" {TEXT 207 26 "por t\351cnicas j\341 aprendida
s" }}{PARA 220 "" 0 "" }{PARA 223 "> " 0 "" {MPLTEXT 1 0 49 "lin_ode_c
onst := subs( p(t)=a, f(t)=b, lin_ode );" }}{PARA 223 "> " 0 "" 
{MPLTEXT 1 0 41 "odeadvisor( lin_ode_const, [separable] );" }}{PARA 
223 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}{PARA 223 ">
 " 0 "" {MPLTEXT 1 0 52 "lin_ode_const_soln := dsolve( lin_ode_const, \+
x(t) );" }}{PARA 223 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 0
:" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 207 102 "Aprendemos em sala de au
la que a solu\347\343o geral de uma EDO linear de primeira ordem \351 \+
a soma da solu\347\343o" }}{PARA 220 "" 0 "" {TEXT 207 44 "geral da eq
ua\347\343o homog\352nea a ela associada (" }{TEXT 210 17 "x\264(t)+p(
t).x(t)=0" }{TEXT 207 40 ") com uma solu\347\343o particular arbitr\34
1ria." }}{PARA 220 "" 0 "" {TEXT 207 26 "No exemplo acima, o termo " }
{TEXT 210 3 "b/a" }{TEXT 207 79 " \351 uma solu\347\343o particular e \+
o termo envolvendo e fun\347\343o exponencial \351 a solu\347\343o" }}
{PARA 220 "" 0 "" {TEXT 207 30 "geral da homogenea associada. " }}}
{EXCHG {PARA 220 "" 0 "" {TEXT 207 102 "A solu\347\343o da homogenea a
ssociada (que \351 de vari\341veis separ\341veis) pode ser obtida e te
stada como abaixo" }{TEXT 207 1 ":" }}{PARA 220 "" 0 "" }{PARA 223 "> \+
" 0 "" {MPLTEXT 1 0 52 "soln_h := x(t) = coeff(rhs(lin_ode_const_soln)
,_C1);" }}{PARA 223 "> " 0 "" {MPLTEXT 1 0 46 "odetest( soln_h, subs( \+
b=0, lin_ode_const ) );" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 207 80 "De \+
forma semelhante, a solu\347\343o particular pode ser obtida e testada
 como abaixo:" }{TEXT 207 1 "\n" }}{PARA 223 "> " 0 "" {MPLTEXT 1 0 
56 "soln_p := x(t) = subs( _C1=0, rhs(lin_ode_const_soln) );" }}{PARA 
223 "> " 0 "" {MPLTEXT 1 0 33 "odetest( soln_p, lin_ode_const );" }}}}
{SECT 0 {PARA 221 "" 0 "4.B" {TEXT 208 19 "3. Fator Integrante" }}
{EXCHG {PARA 220 "" 0 "" {TEXT 207 27 "Sabemos que um dos m\351todos "
 }{TEXT 207 73 "de obten\347\343o de uma solu\347\343o para uma EDO li
near de primeira ordem consiste" }}{PARA 220 "" 0 "" {TEXT 207 29 "em \+
achar um fator integrante " }{XPPEDIT 18 0 "mu(t)" "6#-%#muG6#%\"tG" }
{TEXT -1 1 " " }{TEXT 207 82 "pelo qual a EDO original deve ser multip
licada. Se este fator for convenientemente" }}{PARA 220 "" 0 "" {TEXT 
207 23 "escolhido como abaixo: " }}{PARA 220 "" 0 "" }{PARA 224 "" 0 "
" {TEXT 211 1 " " }{XPPEDIT 18 0 "mu(t) = exp( int( p(t), t ) )" "6#/-
%#muG6#%\"tG-%$expG6#-%$intG6$-%\"pGF&F'" }{TEXT 211 1 "." }}{PARA 
220 "" 0 "" {TEXT 207 37 "a EDO se transforma em outra da forma" }
{XPPEDIT 18 0 "diff( X(t), t ) = F(t)" "6#/-%%diffG6$-%\"XG6#%\"tGF*-%
\"FGF)" }{TEXT 207 2 "\n," }}{PARA 220 "" 0 "" {TEXT 207 5 "onde " }
{XPPEDIT 18 0 "X(t) = mu(t)*x(t)" "6#/-%\"XG6#%\"tG*&-%#muGF&\"\"\"-%
\"xGF&F+" }{TEXT 207 3 " e " }{XPPEDIT 18 0 "F(t) = mu(t)*f(t)" "6#/-%
\"FG6#%\"tG*&-%#muGF&\"\"\"-%\"fGF&F+" }{TEXT 207 59 " , que \351 de r
esolu\347\343o imediata, bastando apenas integr\341-la:" }}{PARA 225 "
" 0 "" {XPPEDIT 18 0 "X(t)" "6#-%\"XG6#%\"tG" }{TEXT 212 3 " = " }
{XPPEDIT 18 0 "Int( diff(X(t), t ), t )" "6#-%$IntG6$-%%diffG6$-%\"XG6
#%\"tGF,F," }{TEXT 212 3 " = " }{XPPEDIT 18 0 "Int( F(t), t ) + C" "6#
,&-%$IntG6$-%\"FG6#%\"tGF*\"\"\"%\"CGF+" }{TEXT 212 1 "." }}{PARA 220 
"" 0 "" {TEXT 207 73 "De posse de X(t), a solu\347\343o desejada pode \+
ser obtida por meio da equa\347\343o " }{XPPEDIT 18 0 "x(t) = X(t)/mu(
t)" "6#/-%\"xG6#%\"tG*&-%\"XGF&\"\"\"-%#muGF&!\"\"" }{TEXT 207 1 "." }
}{PARA 220 "" 0 "" {TEXT 207 25 "Considere a seguinte EDO:" }}{PARA 
223 "> " 0 "" {MPLTEXT 1 0 55 "lin_ode1 := diff( x(t), t ) + x(t)/(t+1
) = ln(t)/(t+1);" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 207 33 "Pode-se cl
aramente observar que  " }{XPPEDIT 18 0 "p(t)=1/(t+1)" "6#/-%\"pG6#%\"
tG*&\"\"\"F),&F'F)F)F)!\"\"" }{TEXT 207 4 " e  " }{XPPEDIT 18 0 "f(t) \+
= ln(t)/(t+1)" "6#/-%\"fG6#%\"tG*&-%#lnGF&\"\"\",&F'F+F+F+!\"\"" }
{TEXT 207 55 ". Assim o fator integrante pode ser calculado da forma:"
 }}{PARA 220 "" 0 "" }{PARA 223 "> " 0 "" {MPLTEXT 1 0 45 "int_fact :=
 mu(t) = exp( Int( 1/(t+1), t ) );" }}{PARA 223 "> " 0 "" {MPLTEXT 1 
0 31 "int_fact1 := value( int_fact );" }}}{EXCHG {PARA 220 "" 0 "" 
{TEXT 207 9 "O pacote " }{HYPERLNK 213 "DEtools" 2 "DEtools" "" }
{TEXT 207 18 " cont\351m o comando " }{HYPERLNK 213 "intfactor" 2 "DEt
ools,intfactor" "" }{TEXT 207 44 ", que permite determinar o fator int
egrante." }}{PARA 220 "" 0 "" }{PARA 223 "> " 0 "" {MPLTEXT 1 0 22 "in
tfactor( lin_ode1 );" }}}{EXCHG {PARA 222 "" 0 "" {TEXT 200 54 "Multip
licando nossa EDO pelo fator integrante, tem-se:" }}{PARA 222 "" 0 "" 
}{PARA 223 "> " 0 "" {MPLTEXT 1 0 42 "ode2 := subs( int_fact1, mu(t)*l
in_ode1 );" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 207 108 "Da forma como e
sta EDO se apresenta, a sua solu\347\343o parece ser bastante complica
da. Entretanto se percebermos" }}{PARA 220 "" 0 "" {TEXT 207 39 "que o
 seu lado direito \351 a derivada de " }{TEXT 210 10 "(t+1).x(t)" }
{TEXT 207 9 ",  temos:" }}{PARA 220 "" 0 "" }{PARA 223 "> " 0 "" 
{MPLTEXT 1 0 61 "ode3 := subs( int_fact1, Diff( mu(t)*x(t), t ) ) = rh
s(ode2);" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 207 50 "Esta EDO pode ser \+
resolvida por integra\347\343o direta:" }}{PARA 220 "" 0 "" }{PARA 
223 "> " 0 "" {MPLTEXT 1 0 31 "int_ode3 := map(Int, ode3, t );" }}}
{EXCHG {PARA 220 "" 0 "" {TEXT 207 64 "Fazendo as substitui\347\365es \+
e a integra\347\343o do lado direito, tem-se:" }}{PARA 220 "" 0 "" }
{PARA 223 "> " 0 "" {MPLTEXT 1 0 62 "q1 := subs( int_fact1, mu(t)*x(t)
 ) = int( rhs(ode3), t ) + C;" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 207 
23 "cuja forma explicita \351:" }}{PARA 220 "" 0 "" }{PARA 223 "> " 0 
"" {MPLTEXT 1 0 36 "expl_soln := solve( q1, \{x(t)\} )[1];" }}}{EXCHG 
{PARA 220 "" 0 "" {TEXT 207 45 "Podemos testar se esta solu\347\343o s
atisfaz a EDO" }}{PARA 220 "" 0 "" }{PARA 223 "> " 0 "" {MPLTEXT 1 0 
31 "odetest( expl_soln, lin_ode1 );" }}}{EXCHG {PARA 220 "" 0 "" 
{TEXT 207 99 "De forma a enfatizar a estrutura da solu\347\343o de uma
 EDO linear em suas partes homogenea e particular" }}{PARA 220 "" 0 ""
 {TEXT 207 34 "repetimos o procedimento j\341 feito:" }}{PARA 220 "" 0
 "" }{PARA 223 "> " 0 "" {MPLTEXT 1 0 44 "soln_h := x(t) = coeff( rhs(
expl_soln), C );" }}{PARA 223 "> " 0 "" {MPLTEXT 1 0 44 "soln_p := x(t
) = subs( C=0, rhs(expl_soln));" }}}{EXCHG {PARA 220 "" 0 "" {TEXT 
207 27 "que pode ser confirmado por" }}{PARA 220 "" 0 "" }{PARA 223 ">
 " 0 "" {MPLTEXT 1 0 35 "odetest( soln_h, lhs(lin_ode1)=0 );" }}{PARA 
223 "> " 0 "" {MPLTEXT 1 0 28 "odetest( soln_p, lin_ode1 );" }}}
{EXCHG {PARA 223 "> " 0 "" }}}{PARA 226 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }