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7/26/2019 trabajo de mate para ing..docx
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Dedicatoria:
o EL PRESENTE TRABAJO MONOGRÁFICO LO DEDICAMOS NUESTROS PADRES POR EL ESFUERZO QUE HACEN DÍA DÍA PARA SACARNOS ADELANTE Y A DIOS POR EL DON DLA VIDA.
INTRODUCCIÓN
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Estud!"#s $% $st$ &!'(tu)# )! t*!%s+#*"!&,% d$ L!')!&$u$ $s u% "/t#d# '!*! !s#&!* ! u%! +u%&,% + #t*! +u%&,% dst%t!- ))!"!d! t*!%s+#*"!d! d$ L!')!&$ d$ +. U%! d$ su'*%&'!)$s 0*tud$s $s u$ t*!%s+#*"! $&u!&#%$d+$*$%&!)$s )%$!)$s $% $&u!&#%$s !)1$2*!&!s- '#* t!%t+3&)$s d$ *$s#)0$*. U%! 0$4 *$su$)t! d&5! $&u!&,!)1$2*!&! s$ 5!))!*3 )! !%tt*!%s+#*"!d! #2t$%/%d#s$ s#)u&,% d$ )! $&u!&,% d+$*$%&!).L! t*!%s+#*"!d! d$ L!')!&$ d$ u%! +u%&,% + 0$%$ d!d! '#"$d# d$ u%! %t$1*!) "'*#'! d$'$%d$%t$ d$ u% '!*3"$t*6)! 0!*!2)$ d$ )! &u!) d$'$%d$ )! +u%&,% F7- '#* )# &u!) t$#*(! $st3 ))$%! d$ &#"')&!&#%$s t/&%&!s. P#* $st! *!4,%- t$%$%d# $% &u$%t! )!s !')&!&#%$s d$ )! t$#*(! u
%$&$st!"#s- '#d$"#s *$st*%1*%#s ! u%! &)!s$ d$ +u%&#%$s$%&))!s- )!s +u%&#%$s d$ #*d$% $9'#%$%&!).E% '*%&'#- 8 d!d# u$ )!s s#)u&#%$s d$ $&u!&#%$d+$*$%&!)$s )%$!)$s &#% &#$:&$%t$s &#%st!%t$s s#+u%&#%$s &#%t%u!s- '#d*(!"#s *$st*%1*%#s ! +u%&#%$&#%t%u!s. S% $"2!*1#- %#s %t$*$s! $stud!* $&u!&#%$s &#"'u)s#s- u$ &#"# 0"#s- 5!&$% u$ )! s#)u&,% &!"2
2*us&!"$%t$ 8 s$! ds&#%t%u!. Es '#* $st# '#* )# ut*!t!*$"#s +u%&#%$s &#%t%u!s ! t*#4#s.
TRANSFORMADA DE LAPLACE
S$! )! +u%&,% d$ F6t7- d$:%t0! ; T ≥0
- )! t*!%s+#*"!d! d$ L!')!&$ d$%#t!'#* L F (t ) # F6s7. S$ d$:%$ '#* )! $&u!&,%<
L [ F (t )]= F ( s)=∫0
∞
e−st
f ( t ) . dt
L [ F (t )]= F ( s)= lim p→ ∞
∫0
p
e−st
. f ( t ) . dt
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T*!%s+#*"!d!s $% 2!s$ ! )! d$:%&,%<
Transformada De Una Funcin Constante
L [1 ]=∫0
p
e−st
.1 . dt
L [1 ]=−15
. e−st
L [1 ]=−0+1
s=
1
s
L [2 ]=2
s L [3 ]=3
s
Transformada De Una Funcin E!"onencia#
L [eat ]=∫0
∞
e−st
. eat
.dt
¿∫0
∞
et (a− s)
. dt
=
1
a−s
.et (a−s)=0−
1
a−s
= 1
s−a
L [e4 t ]= 1
s−4
PROPIEDAD DE LA LINEALIDAD
S$!% )!s +u%&#%$s + 8 1 &#% t*!%s+#*"!d!s d$ L!')!&$ L [ F (t )] 8 L [G(t )] -
*$s'$&t0!"$%t$ 8 s$!% ! 8 2 &#%st!%t$s $%t#%&$s<
L [a .F (t ) ±b.G( t )]=¿ !.L [ F (t )] ± 2.L [G(t )]
E$em"#o
L [e−4 t +10 ]=¿ L [e−4 t ] > [email protected] [1 ]
¿ 1
s+4+10
s
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Transformada de a#%unas funciones&i"er'#icas
L cosh (at ) cosh (at )=1
2 [eat +e
−at ]
L [1
2 (eat +e−at )]=¿ 1
2 .L [eat ] >1
2 .L [e−at ]
¿1
2 . 1
s−a+1
2. 1
s+a
¿1
2 [ s+a+(s−a)
s2−a
2 ]= s
s2−a
2
L [Senh (at )] Senh (at )=1
2 [eat +e−at ]
L [12 (eat −e−at )]=¿
1
2 .L [eat ] 1
2 .L [e−at ]
¿1
2 . 1
s−a−1
2 . 1
s+a
¿1
2 [ s+a−(s−a)
s2−a
2
]= a
s2−a
2
E$em"#o a"#icati(o
L [3.cosh (2 t )−9. senh(8 t )]=¿ .L [cosh (2 t )] −¿ .L [senh (8 t )]
¿3. s
s2−4
−9 8
s2−64
=¿ 3 s
s2−4
− 72
s2−64
Transformada de a#%unas funcionestri%onom)tricas
L cos (wt ) L sen (wt )
eix=cos ( x)+isen( x )
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L e
iwt
[¿ ]=¿ L [cos (wt )+ isen(wt )]
s+iw
s+iw .
1
1−iw=¿ L cos (wt ) −¿ .L sen (wt )
s+iw
s2−(i2 .w
2)=
s+iw
s2+w
2
s
s2+w
2 >iw
s2+w
2=¿ L cos (wt ) −¿ .L sen (wt )
L [sen(wt )]= w
s2+w
2
L [cos (wt )]= s
s2+w
2
E*EMPLO:
L [2.cosh (4 t )−6 senh (3 t )]=¿ .L [cosh (4 t )] −¿ .L [ senh (3 t )]
¿2. 4
e2−16
−6. 3
e2−9
= 8
s2−16
− 18
s2−9
FUNCION +AMMA ,
Γ ¿
Γ (α )=∫0
∞
e− x
. xα −1
. d x
Γ (α +1 )=α . τ (α )
α =n ;n∈ N
Γ (n+1 )=n!
L [ t a ]=∫0
∞
e−st
.t a. dt
Reso#(iendo "or inte%ra# "or "artes-
L [ t a ]= 1
sa+1∫
0
∞
e− x
. xa. dx
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L [ t a ]= τ (α +1 )
sa+1
L [ t a ]= a !
sa+1 a∈ N
L
[ t 3 ]=3 !
s4
TEOREMA DE LA TRANSFORMADA DE LAPLACE DE LAINTE+RAL DE UNA FUNCIÓN
L F (t ) = F (s) - $%t#%&$s L [∫0
t
f (θ ) .dθ ]=¿ F (s)
s
Demostracin:g (t )=∫
0
t
f (θ ) . dθ
g (t )=f (t )
L [ g (t ) ]=s L [g (t )] g(0)
L [ f ( t ) ]=s L [∫0t
f (θ) . dθ
]L
[ f (t )]S =¿
L [∫0
t
f (θ) . dθ]
A"#icando #a transformada in(ersa de
La"#ace F (s )
s =¿ L [∫
0
t
f (θ) . dθ]
L? ( F (s )s )=∫
0
t
f (θ) . dθ
E$em"#o:
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L? [ 1
s(s−4) ]=e4θ
∫0
t
e4θ
dθ=¿ 1
4 e
4θ
=
1
4
e4θ−
1
4
L? [ 1
s2(s−4) ]=∫0
t
[ 14 e4θ−
1
4 ]dθ= 1
16e4θ−
1
4θ
¿ 1
16 e
4 t −1
4 t −
1
16
L? [ 1
sn
]=∫0t
∫0
t
∫0
t
∫0
t
∫0
t
f (θ )dθ
Teor.a de #a inte%ra# de #atransformada de La"#ace
S L f ( t ) = F (s) 8 !d$"3s limt →0 (
f (t )t ) $9st$- $%t#%&$s
L [ f ( t )]=∫s
∞
F (s)d s
Demostracin:
F ( s)=∫s
∞
∫0
∞
e−s t
. f (t ) .dt .
∫s
∞
F ( s) . d s=∫s
∞
∫0
∞
e−s t
. f ( t ) .d t .d s
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¿∫0
∞
[∫s
∞
e−s t
.d s]. f (t ) .dt
¿∫0
∞−1
t . e
−st f ( t ) . dt =∫
0
∞1
t e
−st . f ( t ) . dt
L [ F (t )t ]=∫
s
∞
F ( s) . d s
E$em"#os
P!*! '#d$* !')&!* $st$ "/t#d# t$%$"#s u$ 0$* s $9st$ $) )("t$ d$) + 6t7.
L ["st
t
] C#"'*#2$"#s s $9st$
limt →0
( cos t
t )=1
0=∞ N# $9st$.
L [ sent
t ] C#"'*#2$"#s s $9st$
limt →0
( sin t
t )=0
0
A')&!"#s )! )$8 d$ H,'t!)limt →0
( cos t
1 )=cos0=1 S $9st$
L [ sin t
t ] 1
s2+1 ∫
s
∞
( 1
s2+1 )d s=a#tg( s)
¿ $
2−a#tg (s)
L [2−2cos t
t ] C#"'*#2$"#s s $9st$
limt →0
[2−2cos t
t ]=0
0
A')&!"#s )! )$8 d$ H,'t!)
limt →0
[2 sent
1 ]=2 sent =0 S $9st$
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L [2−2cos t
t ] 2
s +−2 s
s2+1
∫s
∞
(2s +−2 s
s2+1 )d s
2 ln s−ln ( s2+1)
ln( s
s2+1 ) =
ln [ 1
1+ 1
s2 ]=ln 1−ln( s
2
s2+1 )=−ln( s
2
s2+1 )
Teorema de #a trasformada de #a deri(ada-
S % & (t ) $s &#%t%u! ! t*#4#s 8 d$ u% #*d$% $9'#%$%&!) $% u% %t$*0!)#
¿0,+∞ ¿
¿
$%t#%&$s ' { % (t ) =s( (s )− % (0)
D$"#st*!&,%
I%t$1*!d# '#* '!*t$s.
' { % & (t ) }=∫0
∞
e−at
% (t ) dt
+¿ s∫0
∞
e−at
% (t ) dt
¿e−at
% (t ) dt ∫0
∞
¿
¿− % (0 )+s' { % (t )}
¿ s( ( s)− % (0)
C#% u% !*1u"$%t# s")!* '#d$"#s d$"#st*!* u$
' { % & & ( t ) }=− % & (0 )+s' { % & (t )}
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¿− %& (0)+s (S% (s )− % (0 ) )
¿−s2( (s )− %
& (0 )−s%(0)
E$"')#
Us$ $) *$su)t!d# !%t$*#* '!*! &!)&u)!*
¿ '{t 2 }
So#ucin
H!&$%d# f ( t )=t 2
- t$%$"#s u$
' {f (t & )}=s' { f (t ) }−f (0 )
' {2 t }=s' {t 2
}−0
2 ' {t }=s' {t 2 }
2
s2=s' {t 2}
Y !u( &#%&)u"#s u$<
' {t 2 }= 2s2
E) s1u$%t$ *$su)t!d# 1$%$*!)4! )! t*!s+#*"!d! d$ u%! d$*0!d!
T*!s+#*"!d! d$ u%! d$*0!d! 1$%$*!)4!d!
S 86t7-86t7- ..- %2
6t7 s#% &#%t%u!s ! t*#4#s 8 d$ #*d$% $9'#%$%&!) $% $)
%t$*0!)#¿
0,+∞ ¿¿
E%t#%&$s<
' { % & (t ) }=Sn( (s )−∑
)=0
n−1
s) %n−1− )(0)
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¿Sn
( ( s)− %n−1 (0 )−s %
n−2 (0 ) .−sn−1
% (0)
E) s1u$%t$ t$#*$"! t*!t! s#2*$ $) $+$&t# u$ t$%$ )! t*!s+#*"!d! )! $s&!)!&#% d$
u%! +u%&,% f ( t ) .
S$! f ( t ) u%! +u%&,% &#%t%u! ! t*#4#s 8 d$ #*d$% $9'#%$%&!) $%¿
0+∞ ¿
¿
s & * @
D$"#st*!&,% '!*! &#"'*#2!* $st! '*#'$d!d 2!st! d$ u%! &!"2# d$ 0!*!2)$+=t
' {f (t ) }=∫0
∞
e−at
f (t )dt
¿
1
∫0
∞
e
−a+
f (+ ) d+
¿1
F (
s
)
TRANSFORMADA IN/ERSA DELAPLACE
M$d!%t$ )! d$:%&,% d$ )! t*!%s+#*"!d! d$ L!')!&$ s$ t$%$< s F<@-> ∞>→, - $s u%!
+u%&,% s$&&#%!)"$%t$ &#%t%u! 8 d$ #*d$% $9'#%$%&!)- $%t#%&$s ∃ ' { F (t ) }=f (s ) -
A5#*! %0$*t*$"#s $) '*#2)$"!- $s d$&*< d!d! )! +u%&,% +6s7 u$*$"#s $%&#%t*!* )! +u%&,F6t7 u$ &#**$s'#%d$ ! $st! t*!%s+#*"!d! 8 ! $st! +u%&,% F6t7 s$ ))!"! )! t*!%s+#*"!d!
%0$*s! d$ +6s7 8 s$ s"2#)4! '#* '−1
{ f (s)} -$s d$&* F6t7= '−1
{ f (s)} .
E$em"#o:
H!))!* F6t7 s +6s7=2
s+3
So#ucin
F6t7= '−1{ f (s)} = '
−1{ 2
s+3 }=2e−3 t
d$ d#%d$ F6t7= 2e−3 t
EJEMPLO<
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H!))!* F6t7 s +6s7=1
s2+4
F6t7= '−1{ f (s)} = '
−1{ 1
s2+4 } =
1
2 '
−1 { 1
s2+4 } =
1
2 sen2 t K d$ d#%d$ F6t7=
1
2 sen2 t
Pro"iedades de #a transformada in(ersade La"#ace
Primera "ro"iedad de #inea#idadS ! 8 2 s#% &#%st!%t$s !*2t*!*#s 8 +6s7- 16s7 s#% )!s t*!%s+#*"!d! d$ F6t7 8 G6t7*$s'$&t0!"$%t$ $%t#%&$s<
'−1{af ( s)+bg (s)}=a '−1 {f (s)}+b '−1 {g (s)} =! F 6t7 > 2 G 6t7
DemostracinM$d!%t$ )! '*#'$d!d d$ )%$!)d!d d$ )! t*!%s+#*"!d! s$ t$%$<
'−1{af ( s)+bg (s)}=a '
−1 {f (s)}+b '−1 {g (s)} =! F 6t7 > 2 G 6t7
Es d$&* u$< {af (s )+bg(s)}=¿ L {a F (t )+b G(t ) - t#"!%d# )! t*!%s+#*"!d! %0$*s! s$ t$%$<
'−1{af ( s)+bg (s)}=a '
−1 {f (s)}+b '−1 {g (s)} =! F 6t7 > 2 G 6t7
E$"')#<
S +6s7=1
s2 >
1
s2+9
3
s−2 . H!))!* F6t7
So#ucin
F6t7= { f (s ) = '−1{ 1s2 +
1
s2+9
− 3
s−2 } = '−1{ 1s2 } > '
−1{ 1
s2+9 } 3 '
−1{ 1
s−2}
= t >1
3 s$% t e2t
Se%unda "ro"iedad de trans#acin
S '−1{ f (s)} =F6t7K $%t#%&$s '
−1 { f (s−a)} = eat
F6t7
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Demostracin S$ &#%#&$ u$ < s ' {f (t )} =F6s7 → L {eat
F ( t ) } =f (s−a) d$ d#%d$
eat
F6t7= '−1{ f (s−a)} #t*! +#*"! $s<
f (s) = ∫0
+∞
e−st
F (t )dt →f (s−a) = ∫0
+∞
e−(s−a)t
F (t )dt = ∫0
+∞
e−st
. eat F (t )dt = L {eat
F ( t )}
P#* )# t!%t# f (s−a) = L {eat
F ( t )} d$ d#%d$< '−1
{ f (s−a)} = eat
F (t )
E$em"#o: 5!))!* F6t7 s< +6s7 =s−2
(s−2)9+9
F(t)= '−1{ f (s)} = '
−1{ s−2
(s−2)9+9 } = e2t '
−1 { s
s2+9 } = e
2t
&#s t
Tercera "ro"iedad de trans#acin
S '
−1 { f (s ) }=F6t7K $%t#%&$s
'−1{e−at
f ( s ) }= { F (t −a ) - t >a
0 -t <a
Demostracin
C#"# f 6s7 = ∫0
+∞
e−st
F (t ) dt - $%t#%&$s "u)t')&!%d# '#* e−as
e−as
f 6s7 = ∫0
+∞
e−as
. e−st
F (t ) dt = ∫0
+∞
e−s (t +a)
F ( t ) dt
s$! t > ! = u → dt=duK &u!%d# t=@ K u=! 8 &u!%d# t →+∞ K u →+∞
e−as
f 6s7 = ∫0
+∞
e−s (t +a)
F ( t ) dt = ∫0
+∞
e−s+
F (+−a ) d+ = ∫0
+∞
e−s+
F (+−a ) dt
D#%d$ F (+−a )=0
e−as
f 6s7 = ∫0
+∞
e−s+
F (+−a ) dt = ∫0
+∞
e−s t
F (t −a) dt = L { F ( t −a)}
Cuarta "ro"iedad de cam'io de esca#a
S '−1 { f (s ) } =F 6t7K $%t#%&$s '
−1 { f (s ) } =1
F 61
7
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f (s ) = ∫0
+∞
e−s t
F (t ) dt → f (s ) = ∫0
+∞
e−st
F (t ) dt
S$! u=t → dt =d+
d#%d$ t=+
f (s ) 0 ∫0
+∞
e−s t F (t ) dt 0 ∫0
+∞
e−s+ F
(+ )
d+
0 1
∫0
+∞
e−s+
F ( +
) d+
0 1
L { F ( +1
)}
E%t#%&$s< '−1 { f (s ) } =
1
F 61
7
E$em"#o: 5!))!* F6t7 s f 6s7=1
9 s2+1
So#ucin
S$! '−1{ 1
s2+1 } = s$% t →'
−1 { 1
(3 s)2+1 } =1
3 s$%1
3
Transformada in(ersa de La"#ace de #aderi(ada
TEOREMA: S L?6s7 = F6t7 $%t#%&$s L? 6%7 6s7 = 6?7% t% F6t7
Demostracin
C#"# Lt% F6t7 = 6?7% d
n
dsn LF6t7 = 6?7% 6%76s7
T#"!%d# )! %0$*s! ! !"2#s "$"2*#s.
L? 6%76s7 = 6?7% t% F6t7
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E$em"#o- 5!))!* L?)%6 s+2
s+1 7
So#ucin
C#"# Lt F6t7= 6s7 L? 6s7 =t F6t7
L? 6s7 =tL?6s7
Lue%o L? 6s7 =1
t L? 6s7
A')&!%d# $st$ *$su)t!d# !) $$*&&# d!d#.
L?)%6 s+2
s+1 7= L?)%6s>7 )%6s>?7
=
1
t L?
1
s+2
1
s+1
=
1
t 6 e
−2t
e−t
7 =
e−t −e
−2 t
t
Transformada in(ersa de La"#ace de #ainte%ra#
TEOREMA- S L?6s7 = F6t7 $%t#%&$s L? ∫s
∞
/(+)d+ = F (t )
t
Demostracin
C#"# L6s7 = 6s7 L F (t )
t = ∫s
∞
/(+)d+
D$ d#%d$ !) t#"!* )! t*!%s+#*"!d! %0$*s! s$ t$%$.
L? ∫s
∞
/(+)d+ = F (t )
t
E$em"#o- 1 ca#cu#ar #a transformada in(ersa de L? ∫s
∞ds
s2+a
2
So#ucin
S L6t7 = 6s7 L F (t )
t = ∫s
∞
/(s)ds
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De donde L? ∫s
∞
/(s)ds} = F (t )
t
Lue%o si L6s7 = 6t7 L? ∫s
∞
/(s)ds} = F (t )
t
A5#*! !')&!"#s $st$ *$su)t!d# !) $$*&&# d!d#.
L?ds
s2+a
2} =
senat
a L? ∫s
∞ds
s2+a
2 =senat
at
Transformada in(ersa de La"#ace de #amu#ti"#icacin "or s
TEOREMA- 2 s L
?
6s7 = F6t7 8 F6@7 =@ $%t#%&$s L
?
6s7 = F
6t7Demostracin
Como L?6s7 = F6t7 L6t7 = 6s7- d$ d#%d$
L F6t7 = sL F6t7 F6@7 =sL6t7 - $s d$&*<
L F6t7 =s6s7 $%t#%&$s L?s 6s7 = F6t7
E$em"#o- 2 &a##ar L?
1
s+¿¿¿5¿s¿
So#ucin
L?
1
s+¿¿
¿5¿s¿
= e−1
L?
1
s5 } =t 4
e−1
24
L?
1
s+¿¿¿5¿s¿
=t 3e−1
6 t 4
e−1
24 =e−1
24 6tt7
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Transformada in(ersa de La"#ace de #adi(isin "or STEOREMA- S L ? 6s7 = F 6t7 $%t#%&$s <
L ?
/ (S)
s = ∫0
t
F (+ )d+
Demostracin
C#"# L ? 6s7 = F6t7 ⇒ LF6t7 = 6s7 d$ d#%d$
L ∫0
t
F (+ )d+} =/ (s)
s ⇒ L ?
/ (s)s = ∫
0
t
F (+ )d+ }
E$em"#o-2 E%&#%t*!* L ?
1+¿ 1
s2
¿1
s ln ¿
7
So#ucin
L ? ? )%6? >
1+¿ 1
s2
¿}
' 1{1
s ln ¿
= L ? ¿ s
2
¿ln ¿ >?7 ln s2
= 1
t L ? 2 s
s2
2
s = 1
t 6 cos t 7 =
t
1−cos ¿¿2¿¿
E$em"#o-
E%&#%t*!* L ? ? )%? L ?
1+¿ 1
s2
¿1
s ln ¿
7 =
1−cos+¿¿+
2¿¿
∫0
t
¿
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Transformada in(ersa de La"#ace "or e#M)todo de #as Fracciones Parcia#es
L!s +u%&#%$s *!&#%!)$s 0(s)1(s) - d#%d$ P6s7 Y Q6s7 s#% '#)%#"#s $% )!s &u!)$
)#s 1*!d#s d$ P6s7 $s "$%#* u$ $) 1*!d# Q6s7- 'u$d$% $9'*$s!*s$ &#"# u%!su"! d$ +u%&#%$s *!&#%!)$s s"')$s- !')&!%d# $) &*t$*# d$ d$s&#"'#s&,%$stud!d# $% $) &!s# d$ )!s %t$1*!)$s d$ +u%&#%$s *!&#%!)$s.
E$em"#o-2 H!))!* L ? 11s
2−2s+5(s−2)(2s−1)( s+1)
So#ucin
11s2−2s+5
(s−2)(2s−1)( s+1) =
2
s−2 >
3
2 s−1 >
4
s+1 =
2 (2 s−1)(s+1)+3(s−2)( s+1)+4 (s−2)(2s−1)(s−2)(2s−1)( s+1)
?? s 2 2s > = A (2s 2 )7 6s >?7 > B(s 2 2) (s >?7 > C6s - 2) (2s - ?7
?? s2
- 2 s + 5=A (2 s2
> s )7 > B 6 s2
s 7 > C 6 s2
- 5 s + 2)
?? s2
- 2 s + 5 = (2A + B + 2C)
s2
+ (A - B - 5 C) s - A - 2 B + 2C
{ 2 2+3+24 =11 2=5
2−3−54 =−2⇒3=−3
− 2−2 3+24 =54 =2
L ? 11s
2−2 s+5(s−2)(2s−1)( s+1) = L ?
5
S−2 3
2 S−1 >2
s+1
= L ? 1
s−2 3
2 L ?
1
s−1
2> L ?
1
s+1 = e2t 32
e
1
2 > e−1
TEOREMA DE 3EA/ISIDE:S$!% P6s7 8 Q6s7 '#)%#"#s $% )#s &u!)$s P6s7 $s d$ 1*!d# "$%#* u$ $) 1*!d# d$ Q6s7. s Q6s7 t$%$
*!(&$s d+$*$%t$s α 1 - α
2 - α 3 -- α n K $%t#%&$s
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'−1
0(s)1(s) = ∑
=1
n 0(α )1 (α ) e
α . t
Demostracin:
S $) '#)%#"# Q6s7 t$%$ % *!(&$s α 1 - α
2 - α 3 -- α n K '#* )# t!%t# d$ !&u$*d# !) "/t#d# d$
d$s&#"'#s&,% d$ )!s +u%&#%$s *$!&#%$s s$ 'u$d$ $9'*$s!* !s<
0(s)1(s) =
21
s−α 1 > 2
2
s−α 2 > > 2
s−α > >
2n
s−α n 6?7
A )! $&u!&,% )$ "u)t')&!"#s '#* s−α - $s d$&*<
0(s)1(s) 6 s−α 7 = 6
21
s−α 1 > 2
2
s−α 2 > > 2
s−α > >
2n
s−α n 76 s−α 7 67
A5#*! t#"!"#s )"t$ &u!%d# s α 8 !')&!%d# )! *$1)! d$ LH#s't!) - s$ t$%$<
2 = lims α
0(s)1(s)
(s−α ) = lims α
0(s)(s−α )
1(s )
2 = lims α
0(s) . lims α
(s−α )1(s) = lim
s α
0(s) . limsα
0(s). lims α
1
1 5 (s)
2 = 0(α ) .
1
1 (α ) =
0 (α )
1 (α ) $%t#%&$s< 2
=
0 (α )
1 (α ) 67
R$$"')!4!"#s 67 $% 6?7 s$ t$%$<
0(s)1(s) =
0(α 1)1 5 (α 1) .
1
s−α 1
> 0(α 2)
1 5 (α 2) .1
s−α 2
>> 0(α n)
1 5 (α n) .1
s−α n - t#"!%d# )!
%0$*s!
'−1
0(s)
1(s) = '
−1
0(α 1)
1 5 (
α 1)
.1
s−α 1
> 0(α 2)
1 5 (
α 2)
.1
s−α 2
>> 0(α n)
1 5 (
α n)
.1
s−α n
'−1
0(s)1(s) =
0(α 1)1 5 (α
1)e
α 1. t
> 0(α 2)1 5 (α
2)e
α 2. t
> > 0(α n)1 5 (α n)
eα n .t
P#* )# t!%t#: '−1
0(s)1(s) = ∑
=1
n 0(α )1 (α ) e
α . t
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E$em"#o:
Ca#cu#ar: '−1
419 s+37
( s−2 ) (s+1 )(s+3) 5
Q6s7 = 6s76s>?76s>7 = s3
> s2
s
Q6s7 = s2
> s Q67=?- Q6?7=- Q67=?@
P6s7 = ?s W P67=W - P6?7=?X - P67=@
E%t#%&$s< '−1
19 s+37
( s−2 ) (s+1 )(s+3) = 0(2)1 5 (2)
e2 t
> 0(−1)1 5 (−1)
e−t
> 0(−3)1 5 (−3)
e−3 t
= e
2t
e
−t
e
−3t
O'ser(acin:
Su'#%1!"#s u$ +6s7 = 0(s)1(s) s#% '#)%#"#s $% d#%d$ $) 1*!d# P6s7 $s "$%#* u$ $) 1*!d# d$
Q6s7- '$*# $% $st$ &!s# Q6s7 = @ t$%$ u%! *!(4 ! d$ "u)t')&d!d " - "$%t*!s u$ )!s #t*!s *!(&$
b1 - b
2 - b3 - - bn s#% dst%t!s $%t*$ s. E%t#%&$s s$ t$%$<
+6s7 = 0(s)1(s) =
21
(s−a)6 > 2
2
(s−a)6−1 > > 26
(s−a)1 >3
1
s−b1 > >
3n
s−bn
2 = lim 1
( −1 )! .d
ds (s−a)n
+6s7 - = ?----"
E%t#%&$s )! t*!%s+#*"!d! %0$*s! $s<
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'−1
+6s7 = '−1
0(s)1(s) = e
at
6 21t
6−1
( 6−1 )! > 22t
6−2
( 6−2 )! > > 26 7 > 31e
b1t
>
32e
b2t
> > 3n ebn t
LA CON/OLUCIONS$! F 8 G d#s +u%&#%$s &#%t%u!s '#* t*!"#s $% &!d! %t$*0!)# :%t# 8 &$**!d# @ [ t [ 2 8 d$ #*d$$9'#%$%&!). L! +u%&,% u$ d$%#t!*$"#s '#* F\G 8 u$ 0$%$ d!d# '#*<
F6t7 \ G6t7 = ∫0
t
F (+ )G (t −+ ) d+
R$&2$ $) %#"2*$ d$ &#%0#)u&#% d$ )!s +u%&#%$s F 8 G.
E$em"#o:
L! &#%0#)u&#% d$ F6t7= e t 8 G6t7=s$%t $s<
et
\ s$% t = ∫0
t
e+sen (t −+ ) d+ = ∫
0
t
e+(sent"s+−sen+"st )d+
= ∫0
t
e+sent"s+d+ ∫
0
t
e+sen+"std+
= e
+"s+
sent
2 ¿ > e+ sen+¿
e+sen+
"st
2 ¿ e+"s+ ¿ ] ^ t
0
=1
2 et sent"st > e
t sen
2t e
t sent"st > e
t cos
2t ]
1
2 sent > "st
=1
2 et
sent −"st ] $%t#%&$s< et ∗sent =
1
2 et
sent −"st ]
Pro"iedades de #a con(o#ucion:S$!% + 8 1 +u%&#%$s &#%t%u!s $% $) %t$*0!)# @ - >_ - $%t#%&$s<
? + \ 1 = 1 \ + 6)$8 &#%"ut!t0!7 + \ 61 > 57 = +\1 > +\5 6)$8 dst*2ut0!7 6+\1 7\5 = + \ 61\57 6)$8 !s#&!t0!7 +\ @ = @\+ = @
TEOREMA DE LA CON/OLUCION:
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s$!% F6t7 8 G6t7 +u%&#%$s &#%t%u!s '#* t*!"#s ∀ t ` @ 8 d$ #*d$% $9'#%$%&!)- $%t#%&$s<
LF6t7\G6t7 = LF6t7.LG6t7 = +6s7 . 16s7
Demostracin:
S$! +6s7 = LF6t7 = ∫0
∞
e−sα
F ( α ) dα K 16s7 = LG6t7 = ∫0
∞
e−s7
G ( 7 ) d7
+6s7.16s7 = 6 ∫0
∞
e−sα
F ( α ) dα 76 ∫0
∞
e−s7
G ( 7 ) d7 7
+6s7.16s7 = ∫0
∞
∫0
∞
e−s (α + 7 )
F (α ) G ( 7 ) dαd7
+6s7.16s7 = ∫0
∞
F ( α ) dα ∫0
∞
e−s (α + 7 )
G ( 7 )d7
• d$!%d# ! α :# -5!&$"#s ! t= α > 7 $%t#%&$s< dt = d7 d$ "#d# u$ 7 = t α
+6s7.16s7 = ∫0
∞
F ( α ) dα ∫0
∞
e−st
G (t −α ) dt
s
7
= @ t =
α
> @ $%t#%&$s< t =
α
s 7 @ t = α > ∞ $%t#%&$s< t = ∞
+6s7.16s7 = ∫0
∞
e−st
dt ∫0
t
F (α ) G (t −α ) dα =
¿e−st ∫
∫0
∞
¿0¿ t F ( α ) G (t −α ) dα dt = LF \ G
E%t#%&$s< L F6t7 \ G6t7 = LF6t7.LG6t7 = +6s7 . 16s7
TEOREMA DE CON/OLUCION PARA LA TRANSFORMADAIN/ERSA:
Su'#%$%d# u$ '−1
+6s7 = F6t7 8 '−1
16s7 = G6s7
E%t#%&$s<
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'−1
+6s7.16s7 = ∫0
t
F (+ )G (t −+ ) d+ = F \ G d#%d$ F \ G $s )! &#%0#)u&#% d$ F 8 G
Conc#usinL! t*!%s+#*"!d! d$ L!')!&$ $s d$%#"%!d! !s( $% 5#%#* ! P$**$S"#L!')!&$. L! t*!%s+#*"!d! d$ L!')!&$ $s u%! I%t$1*!) I"'*#'!. L! +u%&,Es&!),% U%t!*# t!"2/% $s &#%#&d! &#"# +u%&,% H$!0sd$. A) '*#&$s
%0$*s# d$ $%&#%t*!* + 6t7 ! '!*t* d$ F6s7 s$ )$ &#%#&$ &#"# t*!%s+#*"!d%0$*s! d$ L!')!&$. P!*! &!)&u)!* )! t*!%s+#*"!d! %0$*s! d$ L!')!&$ s$ ut)4)! %t$1*!) d$ B*#"a&5 # %t$1*!) d$ F#u*$*M$))%. L! )%$!)d!d $s u%'*#'$d!d "u8 bt) '!*! *$s#)0$* $&u!&#%$s d+$*$%&!)$s )%$!)$s &#&#$:&$%t$s &#%st!%t$s- ! )! 0$4 '$*"t*3 $) &3)&u)# d$ )! t*!%s+#*"!d! d!)1u%!s +u%&#%$s.
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6I6LIO+RAF7A
5tt'<^^aaa."t8.t$s"."9^$t$^d$'t#s^"^"!
X?^)!')!&$^5#"$.5t"5tt'<^^$u)$*.us.$s^c*$%!t#^&)!s$s^""^)!')!&$.'d+ 5tt'<^^"!t$"!t&!s.u%0!))$.$du.&#^c!*!%1#^B##s^&u*s#
&!'@W.'d+ 5tt'<^^aaa.t$&%u%.$s^!s1%!tu*!s^"$t"!t^T$9t#^E%a$2^
*!%s+#*"!d!d$L!')!&$^T*!%s+#*"!d!d$L!')!&$?.'d+5tt'<^^aaa.t$&d1t!).t&*.!&.&*^*$0st!"!t$"!t&!^&u*s#s
)%$!^E&u!&#%$sD+$*$%&!)$s^EDOG$#^$d#&!'1$#^)!')!&$^%#d$.5t")
5tt'<^^aaa.s&.$5u.$s^s2a$2^$%$*1!s*$%#0!2)$s^MATLAB^s"2#)&#^)!')!&$^)!')!&$.5t")
5tt'<^^'t.a'$d!.#*1^a^T*!%s+#*"!d!d$L!')!&$5tt'<^^5t").*%&#%d$)0!1#.&#"^t*!%s+#*"!d!d$
)!')!&$.5t")5tt'<^^aaa.d"!.u01#.$s^c!u*$!^T*!s+#*"!d!L!')!&$.'d
+ 5tt'<^^%$t)4!"!.us!&5.&)^t$sse@:%!)
e@@@e@@We@e@0$*s#%e@'!*!e@'d+.'d+