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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 31 "A equa\347\343o de diferen\347as linear." }{TEXT 257 0 
"" }}{PARA 256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "" 0 "" {TEXT -1 
17 "George Svetlichny" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res
tart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Definimos a equa\347\343
o de diferen\347as linear y(n+1)=a(n)*y(n)+h(n) na linguagem do Maple.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eq:=y(n+1)=a(n)*y(n)+h(
n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "A t\355tulo de exemplo, es
colhemos o coeficiente " }{TEXT 262 4 "a(n)" }{TEXT -1 25 " e o termo \+
n\343o homog\352neo " }{TEXT 263 4 "h(n)" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a(n):=5;h(n):=2^n+1;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 21 "Resolvemos a equa\347\343o." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(eq,y(n));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 28 "Note a constante arbitr\341ria " }{TEXT 258 5 "
y(0) " }{TEXT -1 117 "que aparece na resposta. O primeiro termo \351 a
 solu\347\343o geral da equa\347\343o homog\352nea, e o resto \351 uma
 solu\347\343o particular." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Vam
os agora impor uma condi\347\343o inicial." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "ci:=y(0)=7;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 
"Resolvemos agora a edo com a condi\347\343o inicial." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rsolve(\{eq,ci\},y(n));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "Agora com um termo n\343o homog\352neo ma
is complicado." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "h(n):=n^3
*exp(-2*n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(eq,y(
n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "h(n):=sin(n);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(eq,y(n));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 12 "Vamos agora " }{TEXT 270 6 "apagar" }{TEXT -1 
15 " as defini\347\365es " }{TEXT 265 4 "a(n)" }{TEXT -1 3 " e " }
{TEXT 267 4 "h(n)" }{TEXT -1 31 " tornando-os de novo genericos." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "unassign('a(n)');unassign('h
(n)');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Resolvemos a equa\347
\343o gen\351rica." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolv
e(eq,y(n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Esta \351 a formul
a apresentado em sala de aula, numa forma um pouco diferente." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Vamos agora escolher " }{TEXT 268 
4 "a(n)" }{TEXT -1 37 " uma constante determinado, diagamos " }{TEXT 
276 3 "5, " }{TEXT -1 1 "e" }{TEXT 277 6 " h(n) " }{TEXT -1 19 " de fo
rma especial." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a(n):=5;h(
n):=n^2*2^n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(eq,y
(n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 275 "Note que o primeiro ter
mo \351 solu\347\343o geral da equa\347\343o homog\352nea e o resto \+
\351 solu\347\343o particular. O segundo termo tamb\351m \351 uma solu
\347\343o da equa\347\343o homog\352nea e portanto podemos considerar \+
os primeiros dois termos como uma outra express\343o para a solu\347
\343o geral da equa\347\343o homog\352nea." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
46 "Os tr\352s \372ltimos termos \351 um polin\363mio de grau " }
{TEXT 278 1 "2" }{TEXT -1 7 " vezes " }{TEXT 279 3 "2^n" }{TEXT -1 61 
" em conformidade com o m\351todo de coeficientes indeterminados." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Vamos agora escolher um termo n
\343o homog\352neo onde o fator exponencial tem base " }{TEXT 280 1 "5
" }{TEXT -1 19 ", o coeficiente de " }{TEXT 281 5 "y(n)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h(n):=n^2*5^n;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "rsolve(eq,y(n));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
82 "Note agora que a solu\347\343o particular, (os tr\352s \372ltimos \+
termos) \351 polin\363mio de grau " }{TEXT 282 0 "" }{TEXT 283 1 "3" }
{TEXT 284 0 "" }{TEXT 285 20 " sem termo constante" }{TEXT 286 0 "" }
{TEXT -1 61 " em conformidade com o m\351todo de coeficientes indeterm
inados." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "O seguinte problema ac
ha a soma das quartas pot\352ncias dos primeiros " }{TEXT 288 1 "n" }
{TEXT -1 10 " inteiros " }{TEXT 289 21 "y(n)=1^4+2^4+...+n^4." }{TEXT 
-1 27 " A equa\347\343o de diferen\347as \351 " }{TEXT 290 19 "y(n+1)=
y(n)+(n+1)^4" }{TEXT -1 14 " e obviamente " }{TEXT 291 6 "y(1)=1" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a(n):=1; h
(n):=(n+1)^4; ci:=y(1)=1; eq;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "rsolve(\{eq,ci\},y(n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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