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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 26 "
Efeito da fun\347\343o de Dirac " }{XPPEDIT 258 0 "delta(t-c);" "6#-%&
deltaG6#,&%\"tG\"\"\"%\"cG!\"\"" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" 
{TEXT -1 3 "por" }}{PARA 256 "" 0 "" {TEXT -1 17 "George Svetlichny" }
}}{EXCHG {PARA 259 "" 0 "" {TEXT 260 114 "O objetivo desta planilha \+
\351 demonstrar o efeito de acrescentar ao termo n\343o homog\352neo d
a equa\347\343o de segunda ordem " }{XPPEDIT 261 0 "`@@`(D,2)(y)(t)+a*
D(y)(t)+b*y(t) = h(t);" "6#/,(---%#@@G6$%\"DG\"\"#6#%\"yG6#%\"tG\"\"\"
*&%\"aGF0--F*6#F-6#F/F0F0*&%\"bGF0-F-6#F/F0F0-%\"hG6#F/" }{TEXT 262 1 
" " }{TEXT -1 1 " " }{TEXT 259 54 "uma \"pancada\" na forma de uma fun
\347\343o impulso de Dirac " }{XPPEDIT 268 0 "delta(t-c);" "6#-%&delta
G6#,&%\"tG\"\"\"%\"cG!\"\"" }{TEXT 263 16 " em algum ponto " }
{XPPEDIT 265 0 "0 <= t;" "6#1\"\"!%\"tG" }{TEXT 266 0 "" }{TEXT 264 1 
" " }{TEXT 267 1 "." }{TEXT 276 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "restart:with(DEtools):with(plots):with(inttrans):" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Definimos agora o procedimento qu
e resolva a equa\347\343o sem e com o termo " }{XPPEDIT 256 0 "delta(t
-c);" "6#-%&deltaG6#,&%\"tG\"\"\"%\"cG!\"\"" }{TEXT -1 152 " indicando
 com cores diferentes a solu\347\343o sem a fun\347\343o impulos, com \+
a fun\347\343o impulso, a posi\347\343o do impulso, e o termo n\343o h
omog\352neo. Neste procedimento " }{TEXT 269 1 "a" }{TEXT -1 3 " e " }
{TEXT 270 1 "b" }{TEXT -1 33 " s\343o os coeficientes na equa\347\343o
, " }{TEXT 271 17 "q=y(0), p=y'(0), " }{TEXT -1 0 "" }{TEXT 272 1 "h" 
}{TEXT -1 26 " \351 o termo n\343o homog\352neo, " }{TEXT 273 1 "c" }
{TEXT -1 28 " \351 o instante do impulso, e " }{TEXT 274 1 "L" }{TEXT 
-1 3 " e " }{TEXT 275 1 "M" }{TEXT -1 36 " sao par\342menros usados na
 plotagem. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "pancada:=pr
oc(a,b,q,p,h,c,L,M) local eq1,eq2,u,v,y1,y2: eq1:=(D@@2)(z)(t)+a*D(z)(
t)+b*z(t)=h(t); eq2:=(D@@2)(z)(t)+a*D(z)(t)+b*z(t)=h(t)+Dirac(t-c);" }
{TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "y1:=dsolve(\{eq1,
z(0)=q,D(z)(0)=p\},z(t)); u(t):=subs(y1,z(t)); y2:=dsolve(\{eq2,z(0)=q
,D(z)(0)=p\},z(t)); v(t):=subs(y2,z(t));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 137 "plot([h(t),[[c,h(c)],[c,h(c)+M]],u(t),v(t)],t=0..L,c
olor=[blue,brown,green,red], legend=[\"h(t)\",\"impulso\",\"sem impuls
o\", \"com impulso\"]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Vamos ver um exemplo t\355pic
o:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "pancada(1,37/4,1,0,t-
>-exp(-t/2)*cos(t),2,6,0.6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "O efeito da \"pacada\" \+
\351 de mudar a derivada  " }{TEXT 280 6 "y'(c) " }{TEXT -1 29 "da sol
u\347\343o no ponto indicado " }{TEXT 281 1 "c" }{TEXT -1 6 " para " }
{TEXT 282 8 "y'(c)+1." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Agora a
presentamos alguns dos exemplos tratados na planilha \"Segunda Ordem L
inear\", primeiro com o termo n\343o homog\352nos zero." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 28 "Duas raizes reais positivas:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "pancada(-3,2,0,-0.1,0,0.5,2,8);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Note que a solu\347\343o sem impu
lso que ia para " }{XPPEDIT 277 0 "-infinity;" "6#,$%)infinityG!\"\"" 
}{TEXT -1 26 ", ap\363s o impulso vai para " }{TEXT 279 1 "+" }{TEXT 
-1 1 " " }{XPPEDIT 278 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 ". " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 48 "Duas raizes reais, uma positiva, outra negativa." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "pancada(1,-2,1,-2,0,1,3,3);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Note que a solu\347\343o que \+
ia decres\347er para zero, ap\363s o impulso cresce para infinito." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 32 "Uma raiz real negativa repetida." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "pancada(1,1/4,0,1,0,2,10,2);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 28 "Raizes complexas conjugadas:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "pancada(1,17/4,0,1,0,4,10,0.6);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 78 "Agora vamos ver os mesmos exemplos com termos n\343o homog\352n
eo diferente de zero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "pa
ncada(-3,2,0,-0.1,t->1-3*t,0.5,2,8);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "pancada(1,-2,1,-2,t->1-1.2*t,1,3,3);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "pancada(1,1/4,0,1,t->0.5*cos(t),2,2
0,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "pancada(1,17/4,0,1
,0.5*sin,4,10,0.6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "panc
ada(1,17/4,0,1,t->1-0.1*t,4,10,0.6);" }}}{EXCHG {PARA 0 "> " 0 "" 
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