trabajo de mate para ing..docx

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



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



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



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 )



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



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:



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



¿∫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 $9st$



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 )}



¿− %& (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)



¿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<



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



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



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



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



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



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

¿



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



'−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



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<



'−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:



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<



'−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.



6I6LIO+RAF7A

5tt'<^âaa."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'<^âaa.t$&%u%.$s^!s1%!tu*!s^"$t"!t^T$9t#Ê%a$2^

*!%s+#*"!d!d$L!')!&$^T*!%s+#*"!d!d$L!')!&$?.'d+5tt'<^âaa.t$&d1t!).t&*.!&.&*^*$0st!"!t$"!t&!^&u*s#s

)%$!Ê&u!&#%$sD+$*$%&!)$sÊDOG$#^$d#&!'1$#^)!')!&$^%#d$.5t")

5tt'<^âaa.s&.$5u.$s^s2a$2^$%$*1!s*$%#0!2)$s^MATLAB^s"2#)&#^)!')!&$^)!')!&$.5t")

5tt'<^^'t.a'$d!.#*1â^T*!%s+#*"!d!d$L!')!&$5tt'<^^5t").*%&#%d$)0!1#.&#"^t*!%s+#*"!d!d$

)!')!&$.5t")5tt'<^âaa.d"!.u01#.$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+

http://www.mty.itesm.mx/etie/deptos/m/ma-841/laplace/home.htm


http://euler.us.es/~renato/clases/mm2/laplace.pdf

http://matematicas.univalle.edu.co/~jarango/Books/curso/cap07.pdf


http://www.tecnun.es/asignaturas/metmat/Texto/En_web/Transformada_de_Laplace/Transformada_de_Laplace_1.pdf


http://www.tec-digital.itcr.ac.cr/revistamatematica/cursos-linea/EcuacionesDiferenciales/EDO-Geo/edo-cap5-geo/laplace/node3.html



http://www.sc.ehu.es/sbweb/energias-renovables/MATLAB/simbolico/laplace/laplace.html


http://pt.wikipedia.org/wiki/Transformada_de_Laplace

http://html.rincondelvago.com/transformada-de-laplace.html


http://www.dma.uvigo.es/~aurea/Trasformada_Laplace.pdf


http://netlizama.usach.cl/tesis%20final%202006%2007%2026%20version%20para%20pdf.pdf


http://euler.us.es/~renato/clases/mm2/laplace.pdf










http://pt.wikipedia.org/wiki/Transformada_de_Laplace









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