Notas Transformada de Laplace

7/26/2019 Notas Transformada de Laplace

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5/83

s

t

V0

V0


6/83


7/83

cosh x= ex+ex

2 sinh x= exex

2 cos x= eix+eix

2 sin x= eixeix

2i

0 sin(t)e

tdt

0 cos(wt)e

tdt

0 sin

2(wt)etdt

0 cos

2(wt)etdt

0 sinh(t)e

2t

dt

0 cosh(t)e

2tdt

0

sin(wt)estdt

w

s

s

w


8/83

a0

estdt

a

s

a

s

|x|< 1

k=0 x

k = 11x

k=0 kx

k = x(1x)2

k=0 e

sk

k=0(1)kesk

k=0 kesk

s

F(s) = s26s+4

s33s2+2s

F(s) = s2+s2(s+1)3

F(s) = s(s+1)3

F(s) = 5s215s11

(s+1)(s2)3

F(s) = 3s22s1

(s3)(s2+1)

F(s) = s2+2s+3

(s2+2s+2)(s2+2s+5)

F(s) = 1s2(s2)

0

tetdt

0 t

2etdt

I=

0

|sin(t)| etdt

I=

k=0

(1)k k+1k

sin(t)etdt.


9/83

f(t)

L{f(t)}= 0

f(s)estdt

f(t)

L{f(t)} f(t) s

L{f(t)}= F(s) L{g(t)}= G(s) L{h(t)}= H(s)

f(t) = 1

L{1} =

0 1 est

dt

= lima

a0

estdt

= lima

1 esas

.

lima

1 esas

s > 0

L{1} = 1s

, s > 0.

f(t) = t

L{t} =0

testdt

= test

s

0

0

e

st

s

dt.

= test

s

0

+1

s

0

estdt.


10/83

test

s

0

lima

te

st

s

a

0

s > 0

1sL{1}

L{t} = 1sL{1}= 1

s2, s > 0.

f(t) = tn

tn1

t2

t3

L{t2} =0

t2estdt

= t2est

s

0

0

2t e

st

s

dt.

= 2

s

0

testdt=2

sL{t}= 2

s

1

s2 =

2

s3, s > 0

L{t3

} =

0 t

3

est

dt

= t3est

s

0

0

3t2 e

st

s

dt.

= 3

s

0

t2estdt=3

sL{t2}= 3

s

2

s3 =

3!

s4, s > 0

tn

L{tn}= n!sn+1

, s > 0.

f(t) = eat

L{eat} = 0

eatestdt

=

0

e(s+a)tdt

= e(s+a)t

s+ a

0

= 1

s + a, s + a > 0

f(t) = sin(wt)

L{sin(wt)} =0

sin(wt)estdt

=

sin(wt)est

s

0 1

s

0

(w cos(wt)) est dt

= w

s

0

cos(wt)estdt

= w

s

cos(wt)e

st

s

0

1s

0

(w sin(wt)) est dt

= w

s

1

s w

s

0

sin(wt)estdt

= w

s2 w

2

s2L{sin(wt)}.


11/83

L{sin(wt)}

L{sin(wt)}= ws2

w2

s2L{sin(wt)}.

L{sin(wt)}1 + w2s2

= ws2

,

L{sin(wt)}= ws2 + w2

, s > 0.

L{f(t)} f(t)

f(t) =

0, 0t45, t > 4.

ba f(t)dt=

xa f(t)dt+

bxf(t)dt

L{f(t)} = 0

f(t)estdt

=

40

f(t)estdt+

4

f(t)estdt

=

40

0 estdt+4

5estdt

= 5est

s

4

=5e4s

s

f(t) = cos(wt)

f(t)

f(t) =

0, 0t31, 3t5,0, t > 5.

f(t) = et2

f(t) = 1t

f(t) c c M > 0 T >0 |f(t)| M ect t > T

f(t) = t2 g(t) = sin(t) h(t) = et

|t2| et, t > 0,

|5 cos(t)| et, t > 2,|et| et, t > 0.

f

g

h


12/83

1

2

3

1

t

et

t2

1

2

3

4

12345

1 2 3

t

et

5 cos(t)

1

2

3

1

t

et

et

f(t) [0, ) c

f(t)

s > c

L{f(t)}=0

f(t)estdt=

T0

f(t)estdt+

T

f(t)estdt.

[0, T] T T c M >0 |f(t)| M ect t > T

T

f(t)estdt

T

|f(t)| estdt

T

Mectestdt

= M

T

e(sc)tdt

= M

c s e(sc)t

T

= M

s c e(sc)T, s > c.

0

f(t)estdt s > c

f(t) F(s) =L{f(t)}

lims

F(s) = 0.

F(s) =

0

f(t)estdt.

u= st

F(s) =1

s

0

fu

s

eudu.


13/83

f

M |f(t)|< M

|F(s)| Ms

0

eudu=M

s .

|F(s)| 0 s

lims

F(s) = 0.

F(s) =L{f(t)} f(t) f(t) =L1{F(s)}

F(s)

f(t)

F(s) t= 0

s > s0 F(s) =G(s)

0

f(t)estdt=

0

g(t)estdt

0

(f(t) g(t))estdt= 0

s > s0 f(t) g(t)

h(t) =

0, 0t 3,

0

h(t)estdt= 0 s > s0.

f(t) g(t)

h(t) =

1, 0t < ,0, t

0<


14/83

F(s) =L{f(t)} f(t) =L1{F(s)}

1

s 1

1

s2 t

1

sn, (n= 1, 2, 3,...)

tn1

(n 1)!

F(s) =L{f(t)} f(t) =L1{F(s)}

a

a2

+s2

sin(at)

1

s+a e

at

s > s0

0

h3(t)estdt =

0

estdt 2

estdt

= est

s

0

est

s2

= 1

s e

s

s e

s

s +

e2s

s = 0, s > 0.

F(s) = 10100+s2 a= 10

L1

10

100 + s2

= sin(10t)

F(s) = 1

s30 n= 30

L1

1

s30

=

t29

29!.

F(s) f(t)

limt

f(t) =L,


15/83

lims0+

sF(s) = L.

sF(s) = s

0

f(t)estdt

= s a

0

f(t)estdt+ sa

f(t)estdt.

a

a

f(t) L limt

f(t) = L

s

a

f(t)estdt sa

Lestdt.

s Ls

esta

=Leas

eas 1 s0 lims

0+sF(s) = L.

F(s) f(t)

c

limt0+

f(t) =L,

lims

sF(s) = L.

sF(s) = s

0

f(t)estdt

= s a0 f(t)e

st

dt+ s ba f(t)e

st

dt+ sb f(t)e

st

dt.

s a

b

f(t) [a, b] |f(t)|< M s

ba

f(t)estdt

s ba

|f(t)|estdt

Ms ba

estdt

Ms

1

sest

b

a

=M(esa

esb).

b

c

M > 0 |f(t)|< M ect t > b s

b

f(t)estdt

sb

|f(t)|estdt

sb

M e(sc)tdt

Ms 1c s e

(sc)t

b

= M s

s c (e(sc)b).


16/83

a

f(t) L limt0

f(t) = L

s

a0

f(t)estdt s a0

Lestdt.

s L

s e

st

a

0=L 1 e

as

eas 0 s lims

sF(s) = L.

t

f(t)

c

k

a)

t

f(t)

c

k

b)

t

f(t)

c c + b

k

c)

t

f(t)

c

k

d)

t

f(t)

c 2c

k

e)

F(s) = ks esc

F(s) =

k

s (1 esc

)

F(s) = ks esc 1 esb

F(s) =kccsesc + 1s2 (e

sc 1)

F(s) =kc2cse

sc 2cse2sc + 1s2

2esc e2sc 1+ 2ks esc e2sc


17/83

f(t) = at

f(t) =

0, 0t


18/83


19/83

L{f(t) + g(t)}= L{f(t)} + L{g(t)}

L{f(t) + g(t)} =0

(f(t) + g(t)) estdt

=

0

f(t)estdt+

0

g(t)estdt

=

0

f(t)estdt +

0

g(t)estdt

= L{f(t)} + L{g(t)} .

f(t) = sin(wt)

eiwt = cos(wt) + i sin(wt)

eiwt = cos(wt) i sin(wt).

sin(wt) = eiwt eiwt

2i

cos(wt) = eiwt + eiwt

2


20/83

L{sin(wt)} = L

eiwt eiwt2i

= 1

2iL

eiwt

1

2iL

eiwt

= 1

2i

1

s iw 1s + iw

= 1

2i

s + iw (s iw)(s iw)(s+ iw)

= 1

2i

2iw

s2 + w2

= w

s2 + w2

f(t) = cos(wt)

f(t) = eat

ebt

f(t) = sinh(wt)

sinh(at) = eat eat

2

cosh(at) = eat + eat

2 .

L{sinh(at)} = Leat eat

2 = 1

2Leat 1

2Leat

= 1

2

1

s a 1

s + a

= 1

2

s+ a (s a)(s a)(s+ a)

= 1

2

2a

s2 a2

= a

s2 a2

f(t) =wt sin(wt)

L{wt sin(wt)} = wL{t} L {sin(wt)}=

w

s2 w

s2 + w2

= w(s2 + w2) ws2

s2(s2 + w2)

= w(s2 + w2) ws2

s2(s2 + w2) =

w3

s2(s2 + w2)


21/83

f(t) = cosh(at)

f(t) = 1cos(wt)

L1 {F(s) + G(s)}= f(t) + g(t).

L1 {F(s) + G(s)} = L1 {L{f(t)} + L{g(t)}}= L1 {L {f(t) + g(t)}}= f(t) + g(t).

F(s) = 2s + 4s2 1s1

L12

s+

4

s2 1

s 1 = 2L11

s+ 4L

1 1s2 L

1 1

s 1= 2 + 4t et

f(t) f(t) t0

L{f(t)}= sL{f(t)} f(0).

f(t) f(t)

L{f(t)} = 0

estf(t)dt

= estf(t)0 0

(sest)f(t)dt

= f(0) + s0

estf(t)dt

= f(0) + sL{f(t)}

f(t) f(t)

f(t) f(t) f(t)

L{f(t)} = sL{f(t)} f(0)= s (sL{f(t)} f(0)) f(0)= s2L{f(t)} sf(0) f(0).

f(t), f(t), , f(n1)(t) f(n)(t)

L{f(n)(t)}= snL{f(t)} sn1f(0) sn2f(0) f(n1)(0).


22/83

f(t) = cos(t)

f(t) = cos(t), f(t) =sin(t) f(t) =cos(t).

f(t) =f(t).

s2F(s) sf(0) f(0) =F(s).

f(0) = cos(0) = 1 f(0) =sin(0) = 0

s2F(s) s=F(s),

F(s) = s

s2 + 1,

g(t) = t cos(t)

g(t) =t sin(t) + cos(t)

g(t) =t cos(t) sin(t) sin(t),

g(t) =g(t) 2 sin(t).

s2G(s) sg(0) g(0) =G(s) 2L{sin(t)}.

g(0) = 0 cos(0) = 0 g(0) =0 sin(0) + cos(0) = 1 L{sin(t)}= 1s2+1

s2G(s) 1 =G(s) 2s2 + 1

,

G(s) = s2 1(s2 + 1)2

,

y(t) + ay(t) + by(t) = f(t)

y(0) = y0

y(0) = y0


23/83

a

b

y0 y0

L{y(t)} + aL{y(t)} + bL{y(t)}=L{f(t)}.

s2L{y(t)} sy(0) y(0) + asL{y(t)} ay(0) + bL{y(t)}=L{f(t)}

Y(s) =F(s) + sy0+ y0+ ay0

(s2 + as+ b)

L{y(t)}= Y(s) L{f(t)}= F(s) y(t) =L1{Y(s)}

y(t) + y(t) = 2t

y(0) = 2

y(0) = 1

F{y(t)

}+

F{y(t)

}=

F{2t

}

s2L{y(t)} sy(0) y(0) + F{y(t)}= 2F{t}. L{y(t)}= Y(s) F{t}= 1s2

s2Y(s) sy(0) y(0) + Y(s) = 2s2

.

y(0) = 2 y(0) = 1

s2Y(s) 2s 1 + Y(s) = 2s2

.

Y(s)

Y(s)

s2 + 1

= 2

s2+ 2s + 1,

Y(s) = 2

s2 (s2 + 1)+

2s

(s2 + 1)+

1

(s2 + 1).

y(t) =L1

2

s2 (s2 + 1)

+ L1

2s

(s2 + 1)

+ L1

1

(s2 + 1)

.

L1 1

s2 (s2 + 1)

= t sin(t), w= 1 ,

L1

s

(s2 + 1)

= cos(t), w= 1

L1

1

(s2 + 1)

= sin(t), w= 1

y(t) = 2t 2sin(t) + 2 cos(t) + sin(t) = 2t + 2 cos(t) sin(t)


24/83

y(t) + 2y(t) + y(t) = et

y(0) = 1y(0) = 1

P(x) Q(x) P Q Q(x)

Q(x) = (a1x + b1)l1 (anx + bn)ln(c1x2 + d1x + e1)p1 (cmx2 + dmx + em)pm .

A1,1, . . . , An,l1ln B1,1, . . . , Bm,p1pn C1,1, . . . , C m,p1pn

P(x)

Q(x) =

l11k=0

A1,k(a1x + b1)l1k

+ +ln1k=0

An,k(anx + bn)lnk

+p11k=0

B1,kx + C1,k(c1x2 + d1x + e1)p1k

+ + pm1k=0

Bm,kx + Cm,k(cmx2 + dmx + em)pmk

.

F(s) = 1(s1)(s22s5)

A

B

C

F(s) = 1

(s 1)(s2 2s 5) = A

s 1+ B+ Cs

s2 2s 5=

A(s2 2s 5) + (B+ Cs)(s 1)(s 1)(s2 2s 5)

=

s2(A + C) + s(

2A + B

C)

5A

B

(s 1)(s2 2s 5) .

A +C = 0

2A + B C = 05A B = 1

A= B =16 C= 16

F(s) =1

6

1

s 1+ 1 + ss2 2s 5

.

F(s) =1

6

1

s 1+ s 1

(s 1)2 6

.

f(t) =1

6

et + et cosh(

6t)

.

F(s) = 3s2 2s 1(s 3)(s2 + 1) .


25/83

f(t) =8t + 2 sin(5t)

f(t) =3t2(t 2)

f(t) = (2t 3)3

f(t) = cos(t)2

f(t) = sin(t)3

f(t) = cosh(t)3

F(s) =8s2 + 10s2+25

F(s) =18s4 + 12s3

F(s) = 48s4 72s3 + 54s2 27s

F(s) = s2+2

s3+4s

F(s) = 6s4+10s2+9

F(s) = s(s27)

s410s2+9

F(s) = 2s8s2+4

F(s) = 2s

1s24 +

1s(s24)

f(t) = 2 cos(2t) 4 sin(2t)

f(t) = 2 cosh(2t)

sinh(2t)

L{ezt}= 1sz z C

L{e(a+ib)t}= s a+ ib(s a)2 + b2


26/83

L{eat cos(bt)}= s a(s a)2 + b2

L{eat sin(bt)}= b(s

a)2 + b2

y+ 2y= 1,y(0) = 3

y+ 3y= et,y(0) = 2

y+ 5y+ 6y= 0,y(0) = 1,y(0) = 0

y

+ 2y+ y= et,y(0) =1,y(0) = 1

y(t) = 52e2t + 12

y(t) = 14et + 74e

3t

y(t) = 3e2t 2e3t

y(t) = 12et(t2 2)

0

1

2

0 1 2 3 4

t

f(t)a)

t

f(t)b)

F(s) = 1s + ese2s

s2

F(s) = 2s +es+2e3se4s

s2


27/83

s

F(sa)

F(s)

F(s) f(t) eatf(t) F(s a)

Leatf(t)= F(s a), s > a

L1 {F(s a)}= eatf(t), s > a.

F(s a)

F(s a) =0

f(t)e(sa)tdt

=0

f(t)eatestdt

= Leatf(t)

f(t) = teat

L{t}= 1s2

=F(s),

f(t) = t

Leatt= F(s a) = 1(s a)2 , s > a

L

1

4a3(sin(at) cosh(at) cos(at) sinh(at))

=

1

s4 + 4a4,

L

1

asin(at)

=

1

s2 + a2,


28/83

L{cos(at)}= ss2 + a2

.

L

1

4a3(sin(at) cosh(at) cos(at) sinh(at))

= L

1

4a3

sin(at)

eat +eat

2 cos(at)e

at eat2

= L 18a3

eat (sin(at) cos(at)) +eat (sin(at) + cos(at))

= 1

8a3Leat (sin(at) cos(at)) + Leat (sin(at) + cos(at))

= 1

8a3

a

(s a)2 +a2 s a

(s a)2 +a2

+ a

(s+a)2 +a2+

s+a

(s+a)2 +a2

= 1

8a3

8a3

((s a)2 +a2) ((s+a)2 +a2)

=

1

s4 + 4a4

L

1

(n 1)! tn1eat

L

1

wsin(wt)eat

f(t) = et cosh(2t)

a= 2

L{cosh(2t)}= ss2 4=F(s).

s

Let cosh(2t)= F(s+ 1) = s + 1

(s+ 1)2

4, s >

1

L{sinh(2t) cos(t)}

L(4 + t2)et

F(s) = 1s22s3 F(s)

F(s) = 1

s2 2s 3= 1

(s 1)2 1 3= 1

(s 1)2 4 .

L12

sinh(2t)

= 1s2 4=G(s)

s

L1{F(s)}=L1{G(s 1)}= 12

et sinh(2t).

f(t) F(s)

F(s) = s(s+1)2+1

F(s) = s1(s22s+5)(s22s+10)


29/83

m

k

y(t)0

y(t)

t

ma=i

fi,

a = y (t)

ifi

k

f1 =ky(t)

f2 =y (t)

f3(t) =f(t)

my(t) = f1+ f2+ f3 =ky(t) y (t) + f(t),

y(t)

my(t) + y (t) + ky(t) = f(t).

y(0) =y0 y(0) =y0

mL{y(t)} + L{y(t)} + kL{y(t)}=L{f(t)}.

ms2

L{y(t)

} msy(0)

my(0) + s

L{y(t)

} y(0) + k

L{y(t)

}=

L{f(t)

}.

ms2Y(s) msy0 my0+ sY(s) y0+ kY(s) = F(s),

F(s) =L{f(t)} Y(s) =L{y(t)} Y(s)

Y(s) =F(s) + msy0+my0+ y0

ms2 + s + k .

y(t) =L1{Y(s)}


30/83

f(t) = 0 F(s) = 0

Y(s) =msy0+ my

0+ y0

ms2 + s + k .

2 0

2 >4mk F(s) = 0

Y(s) =msy0+ my

0+ y0

ms2 + s + k =

msy0+ my0+y0

m s + 2m2 2

4m+ k,

24m +k < 0

2 sinh(at) = eateat 2 cosh(at) =eat + eat

2 = 4mk F(s) = 0

Y(s) =msy0+ my0+ y0

ms2 + s + k =

msy0+ my0+y0s 2m

2 ,

= 0 F(s) = 0

Y(s) = msy0+ my

0

ms2 + k =

1

m

msy0+ my0

s2 + km,

= 2 m= 1 k= 5

y0 = 1 y0 =2 Y(s)

Y(s) = s

s2

+ 2s + 5

.

y(t) =L1

s

s2 + 2s + 5

.

ss2+2s+5

s

s2 + 2s + 5=

s

(s+ 1)2 + 4 =

s + 1

(s + 1)2 + 4 1

(s + 1)2 + 4.


31/83

y(t) = L1

s + 1

(s+ 1)2 + 22

1

2L1

2

(s+ 1)2 + 22

= et cos(2t) 12

et sin(2t).

et

cos(2t) 12

sin(2t)

= Aet cos(2t + ) = Aet (cos(2t)cos() sin(2t)sin()) .

A cos() = 1

A sin() = 1

2

A= 1 +1

4=

5

2

cos() = 2

5=

2

5

5 ,

0, 463648 26, 570

y(t) =

5

2 et cos(2t + 0, 463648).

y(t)

1/2

u(t) =

0, t < 01, t > 0.

t= a

u(t a) =

0, t < a1, t > a.

u(t) u(t

a) a >0

t = a

t = a

t = 0

g(t) =

0, t ,


32/83

t

y(t)

52

et

52

et

1

t

u(t)

1

t

u(t a)

a

1

t

u(t)

g(t) t= 0

lim0

g(t) = u(t), t= 0.


33/83

fp(t) =

0, t < a1, a < t < b0, t > b.,

a < b

1

t

a b

u(t)

fp(t) = u(t a) u(t b), a < b.

fp(t)f(t) f

t < a

f

t = a t = b u(t a)f(t) t= a u(t 1)sin(t)

1

1 2 3 4 5 6 7t

u(t 1) sin(t)

u(t 1) sin(t)

f(t)

f(t) =

0, t < 1

2, 1< t < 33, 3< t < 50, t > 5.

2(u(t 1) u(t 3)) 3(u(t 3) u(t 5))

f(t) = 2(u(t 1) u(t 3)) 3(u(t 3) u(t 5)).

f(t) = (u(t) u(2)) sin(t)


34/83

1

2

1

2

3

1 2 3 4 5 6 7t

f(t)

a0

L{u(t a)} =0

u(t a)estdt

=

a

estdt

= 1

s est

a

=eas

s .

a 0

L1 easF(s)= u(t a)f(t a), a > 0.

L{u(t a)f(t a)} =

0

u(t a)f(t a)estdt

=

a

0

u(t a)f(t a)estdt +a

u(t a)f(t a)estdt

=

a

f(t a)estdt,

u(ta) [0, a) (a, ) v= ta

a

f(t a)estdt=0

f(v)es(v+a)dv= eas0

f(v)esvdv.


35/83

L{u(t a)f(t a)}= easL{f(t)}= easF(s).

f(t) = 1

L{u(t a)}= eass

, a > 0

a= 0

L{t2}= 2s3

e3s 2s3

L1

e3s2

s3

= u(t 3)(t 3)2.

G(s) =e2s ss2+4

G(s) =es 1s2 3e3s 1s2

F(s) = es 1

(s + 1)2 1 .

1(s+1)21

L1

1

(s + 1)2 1

= et sinh(t).

L1

es 1

(s + 1)2 1

= u(t 1)L1

1

(s+ 1)2 1

= u(t 1)e(t1) sinh(t 1).

F(s) =e2s(s 1)s2 2s + 5 .

F(s) f(t) t0

f()d 1sF(s)

L t

0

f()d

=

1

sF(s),

L1

1

sF(s)

=

t0

f()d.


36/83

g(t) =t0

f()d g(t) = f(t)

L{g(t)}= sL{g(t)} g(0).

g(0) = 0

L t

0

f()d = L {g(t)}=

1

sL{g(t)}

= 1

sL{f(t)}

= 1

sF(s).

V0

V(t)

V(t) = V0(u(t a) u(t b)) .

t= a V0 t= a t= b

CR

V

i(t)

Ri(t) + 1

C

q(t) = V(t),

q(t) 1Cq(t) C Ri(t)

R

q(t) =t0

i()d i(t)

Ri(t) + 1

C

t0

i()d =V0(u(t a) u(t b)) .

L{i(t)} + 1sRC

L{i(t)}= V0Rs

eas ebs ,


37/83


38/83

1

2

3

11 2 3 4 51

t

i(t)

1

2

3

11 2 3 4 51

t

q(t)

d1 = 0 d2 = 1CR

L1

1

s

s+ 1CR

= CR

1 e tCR

.

t

q(t) = CV0 u(t a)1 e ta

CR u(t b)1 e tb

CR .

q(t) =

0, t < a

CV0

1 e taCR

, a < t < b,

CV0

e

tb

CR e taCR

, t > b,

a= 0.5 b= 2 R= 1 C= 1 V0 = 3V


39/83

f(t) = u(t

1) + 3u(t

3)

4u(t

5)

f(t) = tu(t) + (t2 t)u(t 1) + (6 t t2)u(t 2) + (t 6)u(t 6)

1

2

3

4

1 2 3 4 5 6

t

f(t)

1

2

3

4

1 2 3 4 5 6

t

f(t)

f(t) = 1s

es + 3e3s 4e5s

f(t) = 1s2 +2s3 +

1s2

es +

2s3 5s2 e2s + 1s2 e6s


40/83

1

2

1

2

1 2 3 4 5 6t

f(t)

1

2

1

2

1 2 3 4 5 6

t

f(t)

1

1 2 3 4 5 6

t

f(t)

1

2

3

4

1 2 3 4 5 6

t

f(t)

f(t) = u(t 1) 3u(t 3) + 2u(t 5)

f(t) = tu(t) + (1 t)u(t 1) + 32 t+ 92u(t 3) + 32 t 112 u(t 5)

f(t) =

n=0(1)nu(t n)


41/83

f(t) =

n=0 u(t 2n)

F(s) = 1s

es 3e3s + 2e5s

F(s) = 1s2

1 es 32e3s + 32e5s + 2se5s

F(s) = 1s(1+es)

F(s) = 1s(1e2s)

F(s)

F(s) = 2s

s2 + 2s + 5

F(s) = 2s 2

(s2 2s + 5)(s2 2s + 10)

F(s) = 2e4s

s2 4

F(s) = 2 e2s

(s 3)(s 1)

F(s) = 2s

s2 + 2s + 5es

F(s) =

2s

2

(s2 2s + 5)(s2 2s + 10)e3s

f(t) = (2 cos(2t) + sin(2t)) et

f(t) = et

5(cos(2t) sin(3t))

f(t) = u(t 4) sinh(2(t 4))

f(t) = u(t 2)

e3(t2) e(t2)

f(t) = u(t 1) (2 cos(2(t 1)) + sin(2(t 1)))

f(t) = u(t 3)

5 (cos(2(t 3)) sin(3(t 3)))

y+ 5y+ 6y= 12et,y(0) = 1,y(0) =1


42/83

y+ 5y+ 6y= 6u(t 3),y(0) = 1,y(0) =1

y(t) =2et

+ e

t

+ 2e3t

y(t) = u(t)

2e2t e3t+ u(t 3) 1 3e2t+6 2e3t+9

f(t) = tu(t 1)

f(t) = t2u(t 1)

F(s) = 1s2 +

1s e

s

F(s) = 2s3 +

2s2 +

1s

es

F(t) = F0sin

km t

my(t) + ky(t) = F(t)y(0) = 0y(0) = 0

.

y(t) = F0

2k

sin

km

t

km

t cos

km

t

F(t) = 2(u(t 1) u(t 2))

y(t) 3y(t) + 2y(t) = F(t)y(0) = 0y(0) = 0

.

y(t) = (1

2e(t1) + e2(t1))u(t

1)

(1

2e(t2) + e2(t2))u(t

2)


43/83

[, ]

(t) = 1

2(u(t + ) u(t )) =

0, t .

t= a

(t a) = 12

(u(t (a )) u(t (a+ ))) =

0, t < a 12 , a < t < a + 0, t > a + .

(t a) = 1 > 0 (t a) a > 0

= 1 = 12 = 14

= 18 = 112

(t a) = lim0

(t a).

(t a) (t a)

(t)

f(t) = 1

e

t2

2


44/83

t

(t a) a= 1 = 1

a

t

(t a) a= 1 = 12

a

t

(t a) a= 1 = 14

a

t

(t a) a= 1 = 18

a

t

(t a) a= 1 = 112

a

t

(t a) a= 1

a

t= a

(t a) =

0, t=a, t= a

(t a)dt= 1

f(t) t= a

(t a)f(t)dt= f(a).


45/83

V0

t= a [a, a+] f(t) f(a)

(t a)f(t)dt =

(t a)f(t)dt

(t

a)f(a)dt

= f(a)

(t a)dt= f(a).

(t a) = lim0

1

2(u(t (a )) u(t (a + ))) .

lim0

1

2(u((t a) + )) u((t a) ))) = d

dtu(t a)

(t a) = ddt

u(t a).

L{(t a)}=0

(t a)estdt= eas.

V0

V(t)

V(t) = V0(u(t a) u(t b)) .

i(t)

Li(t) + Ri(t) + 1

Cq(t) = V0(u(t a) u(t b)) ,

q(t) 1Cq(t) C Ri(t)

R

Li(t) L i(0) = 0 q(0) = 0

dq(t)dt =i(t)

Li(t) + Ri(t) + 1

Ci(t) = V0((t a) (t b)) ,

i(0) = 0 i(0) = 0

Ls2I(s) + RsI(s) + 1

CI(s) =V0

eas ebs ,


46/83

L

CR

V

I(s) = V0

eas

ebs

Ls2 + Rs + 1C

= 1

L

V0

eas ebss+ R2L

2 R2L2 + 1LC=

V0L

eas

s + R2L2

+ e

bss + R2L

2+

= 1

LC

R

2L

2.

V0 = 10V

a= 1

b= 5

>0 L= 1 C= 110 R= 2

I(s) = 10

es

(s+ 1)2

+ 9 e

5s

(s+ 1)2

+ 9

.

i(t) =10

3

u(t 1)e(t1) sin(3(t 1)) u(t 5)e(t5) sin (3(t 5))

.


47/83

V0

1

2

3

1

2

1 2 3 4 5 6 7 81t

i(t)

1

2

3

4

5

11 2 3 4 5 6 7 81

t

i(t)

= 0 L= 1 C= 1 R= 2

I(s) = 10

es

(s+ 1)2

e5s

(s+ 1)2

.

i(t) = 10

u(t 1)e(t1)(t 1) u(t 5)e(t5)(t 5)

.


48/83

1 2 3 4 5 6 7 81

t

i(t)

L

x

x = L

W(x) x[0, L]

k(x) (x) (x)

x

k(x) =d(x)

dx .

dy(x)

dx = tan((x))

(x) tan((x))(x) dy(x)

dx =(x),

d2y(x)

dx2 =k(x).

k(x) = M(x)EI E I

M(x)

d2y(x)dx2

= M(x)EI

.

M(x) V(x)

d

dxM(x) = V(x)

d

dxV(x) = W(x).


49/83


50/83

1

2

1 1 2 3 4 5 x

y(x)

c(t) +1

c(t) =x(t), t > 0

c(t) x(t) t

c0 T

x(t) =c0((t) + (t T) + (t 2T) + (t 3T) + ) c(0) = 0 c(t)

sC(s) +1

C(s) = c0

1 + esT + e2sT + e3sT + = c0

n=0

esT

n.

C(s)

s+ 1

= c0

n=0

esT

n

C(s) =

c0s+ 1

n=0

esT

n .

s

c(t) = c0

e

t

+ etT

u(t T) + e t2T u(t 2T) + e t3T u(t 3T) +

= c0e t

1 + e

T

u(t T) + e 2T u(t 2T) + e 3T u(t 3T) +

c0 = 1 = 1 T = 1


51/83

1

2

3

1 2 3 4 5

t

i(t)

c0

limtnT

c(t) = limtnT

c0e

t

1 + e

T

+ e2T + + e (n1)T

=

c0enT

1 + e

T

+ e

2T

+ + e(n1)T

=

c0

e

nT

+ e(n1)T

+ e(n2)T

+ + eT

limtnT+

c(t) = limtnT+

c0e

t

1 + e

T

+ e2T + + e (n1)T + enT

=

c0enT

1 + e

T

+ e2T + + e (n1)T + enT

=

c0

e

nT

+ e(n1)T

+ e(n2)T

+ + eT + 1

,

c0

t = 0 c(0) = c0 c(0) = 0

y(t) + y(t) = (t)

y(0) = 0

sY(s) y(0) + Y(s) = 1

Y(s) = 1s+1

y(t) = et.

y(0) = 1= 0

t > 0 y(t)

y(t) = u(t)et.

y(0)

y(t) + y(t) = u(t)u(t)y(0) = 0


52/83

u(t) u(t )

(t) 0+ Y(s)

Y(s) =

1

s(s+ 1)

1

es

=1

s 1

s+ 11

es

,

y(t) =1 et

u(t) u(t ) 1 e

(t)

.

y(t) =

0, t0,

1et

, 0< t,e1

et, t.

>0

et = 1 + t+t2

2 +

t3

3!+ . . .1 + t

y(t)

0, t0,t , 0< t,

et, t. 0 y(t)

y(t) = 0.2 = 0.1 = 0.05

etu(t)

1

0 1 2

[0, ] f g f g

(f g)(t) = t0

f()g(t )d.


53/83

f(t) = et g(t) = cos(t) f g

(f g)(t) = t0

e cos(t )d

= 1

2e (cos(t ) sin(t ))

t

0

= 1

2

et cos(t) + sin(t) .

e cos(t )d= 12e (cos(t ) sin(t )) +

f g= g f

F(s) =L{f(t)} G(s) =L{g(t)}

L{(f g)(t)}= F(s)G(s).

L1

{F(s)G(s)

}= (f

g)(t).

F(s) =L{f(t)}=0

f(t)estdt

G(s) =L{g()}=0

g()esd.

F(s)G(s) =

0

f(t)estdt

0

g()esd

=

0

f(t)

0

g()es(t+)ddt

t

v= t+

F(s)G(s) =

0

f(t)

t

g(v t)esvdvdt

v [t, ] t [0, ] t [0, v] v [0, ]

F(s)G(s) =0

v

0

f(t)g(v t)esvdtdv

=

0

v0

f(t)g(v t)dt

esvdv

=

0

(f g)esvdv= L{f g}


54/83

s(s1)(s2+1)

s

(s 1)(s2 + 1) = 1

s 1s

s2 + 1,

L1 1

s1= et L1 ss2+1= cos(t) L1

1

s 1s

s2 + 1

=

t0

e cos(t )d.

L1

1

s 1s

s2 + 1

=

1

2

et cos(t) + sin(t) .

y(t) = 4 + 9

t0

y()(t )d.

f(t) = y(t) g(t) = t

L{y(t)}= 4s

+ 9L{y(t)}L{t}

Y(s) = 4

s+ 9Y(s)

1

s2.

Y(s) = s2

s2 94

s = 4s

s2 9 .

y(t) = 4 cosh(3t)

y(t) = t +

t0

y()sin(t )d

du(t)

dt =(u(t) uamb) + q(t)

u(t) uamb q(t) u(0) = 50 uamb= 50 = 4

u(t) q(t) = 100(t 1) u(t)


55/83

q(t)

dq(t)

dt =(ua u(t)).

ua

f(t) g(t)

f(t) =

t2

, 0t < 2t2 ,

t


56/83


57/83

f(t) T

L{f(t)}= 11 esT

T0

f(t)estdt.

f(t)

L{f(t)} =0

f(t)estdt

= T

0

f(t)estdt+ 2T

T

f(t)estdt+ 4T

3T

f(t)estdt+

=

n=0

(n+1)TnT

f(t)estdt.

=t nT

L{f(t)} =n=0

T0

f(+ nT)es(+nT)d

=

n=0

esnT T0

f(+ nT)esd.

f() =f(+ nT)

L{f(t)} = n=0

esnT T

0

f()esd

=

T0

f()esd

n=0

esT

n

=

T0

f()esd

1

1 esT

= 1

1 esT T0

f()esd,


58/83

esT

f(t) = cos(wt) cos(wt)estdt=

est (w sin(wt) s cos(wt))s2 + w2

+

L{cos(wt)} = 11 es 2w

2w

0

cos(wt)estdt

= 1

1 es 2w

est (w sin(wt) s cos(wt))s2 + w2

2w

0

= 1

1 es 2ws ses 2w

s2 + w2

= s

s2 + w2.

f(t) 2a

0

1

1

t

f(t)

a 2a 3a 4a 5a 6a

T= 2a

L{f(t)} = 11 e2sa

2a0

f(t)estdt

= 1

1 e2sa a

0

estdt 2aa

estdt

= 1

1 e2sa

1 2eas + e2ass

= 1

(1 esa)(1 + esa)

(1 eas)2s

= 1

s

1 eas1 + esa

.

eas

2

L{f(t)} = 1s

eas

2 e as2eas

2 + eas

2 )

= 1

s

sinhas2

cosh

as2

=

1

stanh

as2

.


59/83

g(t) 2a

0

1

t

g(t)

2a 4a 6a

g(t) [0, a] 1a [a, 2a] 1a

1

a

L{g(t)}= 1sL{g(t)} +1

sg(0).

L{ } = 1as

L{ } +1s

,

L{g(t)} = 1

as

1

stanhas

2

= 1

as2tanh

as2

.

h(t)

h(t) =

sin(wt), 0< t < w0, w < t 0

L

2

t(1 cos(wt))

= ln

s2 + w2

s2

.

L(1

cos(wt)) =

1

s s

s2 + w2.

L

2

t(1 cos(wt))

= 2

s

1

v v

v2 + w2

dv

= 2

ln(v) 1

2ln(v2 + w2)

s

= (2 l n(v) ln(v b))|s= ln

v2

v2 + w2

s

= ln s2

s2 +w2

= ln

s2 + w2

s2

limv

ln

v2

v2 + w2

= ln

limv

v2

v2 + w2

= ln ( 1)

= 0.

ln(x) x >0

L

1

tsin(wt)

= arctan

ws

.

L{sin(wt)}= ws2 + w2

.


68/83

L

1

tsin(wt)

=

s

w

v2 + w2

dv

= arctan v

w

s

= limv arctan v

w arctan sw

=

2 arctan

sw

tan2

= cot()

L

1

tsin(wt)

= cot1

sw

.

= cot1sw

cot = sw

1tan =

sw tan =

ws

L1t

sin(wt)

= arctan

ws

L{ (t)}= 1s

cot1(s),

(t)

(t) =

t0

sin(x)

x dx.

L{ (t)}=L t

0

sin(x)

x dx

=

1

sL

sin(t)

t

.

L{ (t)}= 1s

arctan

1

s

=

1

scot1 (s) .

s


69/83


70/83


71/83

40i

+

110

5 i1

1

10 i2

2

i1 i2

i1 i2 i1(0) =i2(0) = 0

di1(t)

dt + 5i1(t) + 40i(t) = 110

2di2(t)

dt + 10i2(t) + 40i(t) = 110.

i(t) =i1(t) + i2(t)

di1(t)dt

+ 45i1(t) + 40i2(t) = 110

2di2(t)

dt + 40i1(t) + 50i2(t) = 110,

di1(t)

dt + 45i1(t) + 40i2(t) = 110

di2(t)

dt + 20i1(t) + 25i2(t) = 55.

sI1(s) i1(0) + 45I1(s) + 40I2(s) = 110s

sI2(s) i2(0) + 20I1(s) + 25I2(s) = 55s

.

(s+ 45) I1(s) + 40I2(s) = 110

s

20I1(s) + (s + 25) I2(s) = 55

s .


72/83

(s + 45) 40

20 (s + 25)

I1(s)I2(s)

=

110s

55s

I1(s)I2(s)

=

1

(s+ 25)(s + 45) 800

(s + 25) 4020 (s + 45)

110s55s

I1(s) = 1

s2 + 70s + 325

110

s (s+ 25) 2200

s

=

1

(s + 5)(s+ 65)

110 +

550

s

I2(s) = 1

s2 + 70s + 325

2200

s +

55

s(s + 45)

=

1

(s+ 5)(s+ 65)

55 +

275

s

.

I1(s) = 2I2(s) I2(s)

I2

= 1

(s+ 5)(s + 65)55s + 275

s= 55

(s+ 5)(s + 65)s + 5

s= 55

s(s+ 65).

i2(t) = 55

65

1 e65t= 11

13

1 e65t .

i1(t) = 2i2(t)

i1(t) =22

13

1 e65t .

k1 k2 m1 m2

m1x1(t) = k1x1(t) + k2[x2(t) x1(t)] + f1(t)m2x2(t) = k2[x2(t) x1(t)] + f2(t),

x1 x2 f1 f2

m1(s2X1(s) x1(0) sx1(0)) = (k1+ k2)X1(s) + k2X2(s) + F1(s)

m2(s2X2(s) x2(0) sx2(0)) = k2X2(s) + k2X1(s) + F2(s)

m1s2 + k1+ k2X1(s) k2X2(s) = F1(s) + m1x1(0) + sm1x1(0)k2X1(s) + m2s2 + k2X2(s) = F2(s) + m2x2(0) + sm2x2(0)

m1s2 + k1+ k2 k2

k2 m2s2 + k2

X1(s)X2(s)

=

F1(s) + m1x1(0) + sm1x1(0)F2(s) + m2x2(0) + sm2x2(0)

X1(s)X2(s)

=

1

P(s)

m2s2 + k2 k2

k2 m1s2 + k1+ k2

F1(s) + m1x1(0) + sm1x1(0)F2(s) + m2x2(0) + sm2x2(0)

,


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t

x(t)

t

y(t)

RST

R

S

T

/l

x(t) y(t) z(t)

x(t) = x(t)y(t) = x(t) y (t)z(t) = y (t),

x(0) = 1, y(0) =z(0) = 0.

x(t) y(t) z(t)

= 1 = 2

sX(s) x(0) = X(s)sY(s) y(0) = X(s) Y(s)sZ(s) z(0) = Y(s).

X(s) = x(0)

s + =

1

s+ 1.


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Y(s) =X(s)

s+ =

x(0)

(s+ )(s + )=

1

(s + 1)(s+ 2).

Z(s) =

Y(s)

s =

2

s(s+ 1)(s + 2) .

x(t) y(t) z(t)

x(t) =L1 {X(s)}= et

y(t) =L1 {Y(s)}=e2t + et,

a= 2 b=1

z(t) =L1 {Z(s)} = 2L1Y(s)

s = 2

t

0

y()d

= 2

t0

e2 + e d= 2

e2t 1

2 et 1

= e2t 2et + 1,

x(t) y(t) z(t)

t

x(t) y(t)

z(t)


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(k) =

0

exxk1dx, (k >0)

I(x) =m=0

(1)mm!(m+ + 1)

x2

2m+

0

J0(x) = 1 x2

22(1!)2+

x4

24(2!)2 x

6

26(3!)2+

(t) =

t0

sin(x)

x dx

= 0.57721566490153286060651209008240243104215933593992...


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L{f(t) +g (t)}= L{f(t)} +L{g(t)}

Lf(t)= sL{f(t)} f(0)Lf(t)= s2L{f(t)} sf(0) f(0)

s Leatf(t)= F(s a) t L {u(t a)f(t a)}= easF(s)

L{u(t a)}= eas

s

L

t

0

f()d

=

F(s)

s

L{(t a)}= eas

L{(f g)(t)}= F(s)G(s),

(f

g)(t) =

t

0

f()g(t

)d

L{f(t)}= 11 esT

T

0

es

f()d

L{tf(t)}=dF(s)ds

Lf(t)

t

=

s

F(s)ds


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Notas Transformada de Laplace

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