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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 268 48 "MAT1154 \+
- Equa\347\365es Diferenciais e de Diferen\347as " }}{PARA 257 "" 0 "
" {TEXT -1 62 "A equa\347\343o diferencial linear de primeira ordem: y
'+a(x)y=h(x)." }{TEXT 269 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 "por" }}
{PARA 256 "" 0 "" {TEXT -1 17 "George Svetlichny" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 23 "restart: with(DEtools):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 66 "Vamos come\347ar com uma equa\347\343o homog\352ne
a com coeficiente constante." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "eq:=D(y)(x)+2*y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "dsolve(eq,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Note a co
nstante arbitr\341ria " }{TEXT 256 3 "_Cl" }{TEXT -1 13 " que agora \+
\351 " }{TEXT 257 15 "multiplicativa." }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 79 "Vamos impor agora uma condi\347\343o inicial, e
 resolver a equa\347\343o com esta condi\347\343o." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 12 "ci:=y(2)=-1;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "dsolve([eq,ci],y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "
Agora uma condi\347\343o inicial gen\351rica." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "ci:=y(a)=b;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "dsolve([eq,ci],y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "
Agora colocamos um termo n\343o homog\352neo." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "eq:=D(y)(x)+2*y(x)=x**2 -3*x+2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(x));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 124 "Note que os primeiros tres termos d\343o uma s
olu\347\343o particular enquanto o \372ltimo termo \351 a solu\347\343
o geral da equa\347\343o homog\352nea." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "dsolve([eq,ci],y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "ci:=y(0)=A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve
([eq,ci],y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eq:=D(y)
(x)+2*y(x)=exp(-x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsol
ve(eq,y(x));" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "eq:=D(y)(x)+2*y(x)=exp(-2*x);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(x));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "eq:=D(y)(x)+a*y(x)=exp(x);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(x));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "Note que quando " }{TEXT 258 4 "a=-1" }{TEXT -1 53 " a re
sposta n\343o faz sentido. Refazemos a equa\347\343o com " }{TEXT 259 
4 "a=-1" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
eq:=D(y)(x)-y(x)=exp(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"dsolve(eq,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Note que se \+
" }{TEXT 260 3 "_C1" }{TEXT -1 35 " \351 uma constante arbitr\341tia, \+
ent\343o " }{TEXT 261 12 "-1/(1+a)+_C1" }{TEXT -1 48 " tamb\351m o \+
\351. Vamos ent\343o escrever a solu\347\343o com " }{TEXT 262 2 "a " 
}{TEXT -1 28 "arbitr\341ria na seginte forma:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 50 "expr:= (exp(x*(1+a))/(1+a)-1/(1+a)+_C1)*exp(-a*x
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "limit(expr, a=-1);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Isto agora coincide com a solu
\347\343o obtida para o caso " }{TEXT 263 4 "a=-1" }{TEXT -1 56 ". Ass
im embora que a express\343o para a solu\347\343o geral com " }{TEXT 
264 2 "a " }{TEXT -1 34 "arbitr\341ria n\343o faz sentido quando " }
{TEXT 265 4 "a=-1" }{TEXT -1 17 ", a solu\347\343o para " }{TEXT 266 
4 "a=-1" }{TEXT -1 3 " \351 " }{TEXT 267 8 "embutida" }{TEXT -1 48 " n
a solu\347\343o geral e pode ser obtido como limite." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "eq:=D(y)(x)+a(x)*y(x)=0;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(x));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "eq:=D(y)(x)+a(x)*y(x)=h(x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(x));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "ci:=y(a)=b;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "dsolve([eq,ci],y(x));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 187 "Note que o Maple ERROU! Deixou a primeira integral como \+
integral indefinida e portanto sem especificar o que deve ser o valor \+
da mesma. Deste jeito n\343o d\341 para acertar a condi\347\343o inici
al." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ci:=y(0)=1;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve([eq,ci],y(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h(x):=x+1;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve([eq,ci],y(x));" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 104 "Note que o Maple ERROU DE NOVO! Agora \351 a int
egral no primeiro expoente que foi deixado como indefinido." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a(x):=x;" }}}{EXCHG {PARA 0 "
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0 "" {TEXT -1 64 "Finalmente a resposta esta certa! Note o uso da fun
\347\343o especial " }{TEXT 270 3 "erf" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(x));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "a(x):=5;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "dsolve([eq,ci],y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
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