Asig de transf_y_fourier avanzado

ASIGNACIÒN DE EJERCICIOS DE LA UNIDAD III: TRANSFORMADA DE LAPLACE Y SERIES DE
FOURIER. VALOR: 10 PUNTOS.

1.- UTILIZAR LA DEFINICION DE TRANSFORMADA DE LAPLACE Y RESOLVER LA SIGUIENTE
FUNCION
5 2
F t t 7 5 cos 3 t
3
5 2 t
LF t t 7 5 cos 3 t e 1t dt
u
3
t 5 2 t t
LF t t e 1t dt e 5t 7 dt 5 cos 3 t e 5t
dt
u 3 u u

Resolviendo la 1era Integral
t 5 2 1t 5 b
t e dt lim b 0 t2 e 1t
dt
u 3 3 0

u t2 dv e 1t dt
1t 1 1t
du 2tdt v e dt e
5
1 1
t2 e 1t
dt t2 e 1t
e 1t
2tdt
5 5
2 1t t 2 1t 2 1t
t e dt e e tdt
5 5
u t dv e 1t dt
1t
du dt v e dt
1 1t
v e
5
2 1t t2 1t 2 1 1t 1 1t
t e dt e t e e dt
5 5 5 5
2 1t t2 1t 2 1t 2 e 1t
t e dt e t e C
5 52 52 5
2 1t t2 1t 2 1t 2 1t
t e dt e t e e C
5 52 53

Sustituyendo
b
t 5 2 1t 5 t2 1t 2 1t 2 1t
t e dt lim b 0 e t e e
u 3 3 5 52 53 0

t 5 2 1t 5 t2 1t 2 5b 2 5b 2 0
t e dt lim b 0 e e e e
u 3 3 5 52 53 53
5 2 1t
t 5 2 2
t e dt 3
u 3 3 5 3 52
Resolviendo la Segunda Integral
b
t
5t
b
5t 1 5t
e 7 dt lim b 7 e dt 7 lim b e
u 0 5 0

t
5t 1 5b 1 0b 1 7
e 7 dt 7 lim b e e 7
u 5 5 5 5
Resolviendo la Tercera Integral
t b
5t 5t
5 cos 3 t e dt 5 lim b e cos 3 t dt
u 0
b
5t
I e cos 3 t dt
0
5t
u cos 3 t dv e dt
1 5t
du 3 sen 3t v e
5
1 5t 1 5t
I cos 3 t e e 3 sen 3t dt
5 5
1 5t 3
I e cos 3 t e 5t sen 3tdt
5 5
u sen 3 t dv e 5t dt
1 5t
du 3 cos 3t v e
5
1 5t 3 1 5t 1 5t
I e cos 3 t sen 3 t e e 3 cos 3t dt
5 5 5 5
1 5t 3 5t 3
I e cos 3 t 2
e sen 3 t I
5 5 52
3 1 5t 3 5t
I 2
I e cos 3 t 2
e sen 3 t C
5 5 5
3 1 5t
1 2 I e 5 cos 3 t 3 sen 3t
5 52
52 3 1 5t
I e 5 cos 3 t 3 sen 3t
52 52
1
I 2
5 cos 3 t 3 sen 3t
5 3

1 b
5 cos 3 te 1t dt 5 lim b 5 cos 3 t 3 sen 3t
0 52 3 0

1t 5
5e cos 3 tdt 2
lim b b cos 3 b 3 sen 3b 5
0 5 3
1t 52
5e cos 3 tdt
0 52 3
2 7 52
LFt
3 52 5 52 3

2.- UTILIZAR PROPIEDADES Y TABLA PARA DETERMINAR LA TRANSFORMADA DE LAPLACE.
ENUNCIE LAS PROPIEDADES ANTES DE RESOLVER. SIMPLIFIQUE LOS RESULTADOS.
7 4t 2
a) F t e ( cos 2 5t 2 cosh2 3t 4t 7 )
2 3
7 4t
F t e cos 2 5t 7e 4t cosh2 3t 14e 4t t 7
3
7 4t
L F t L e cos 2 5t 7e 4t cosh2 3t 14e 4t t 7
3
7
L F t L e 4t cos 2 5t 7 L e 4t cosh2 3t 14 L e 4t t 7
3
7
L F t L cos 2 5t ( S 4) 7 L cosh2 3t ( S 4) 14 L t 7 ( S 4)
3
7 S 4 S 4 7!
L F t 7L 14
3 ( S 4) 2 2 5
2
( S 4) 2 2 3
2
( S 4)8
7 S 4 S 4 70560
L F t 2
7 2
3 ( S 4) 20 ( S 4) 12 ( S 4)8
7( S 4) 7 S ( 4) 70560
LF t 2 2
3( S 8S 36) S 8S 28 ( S 4)8

3 sen3t
b) F t t 6 senh 2t 5 2
5 t
18 sen3t
Ft t senh 2t 3
5 t
18 sen3t
LFt L t senh 2t 3
5 t

18 sen3t
LFt L t senh2t 3L
5 t
18 d 2
LFt ( 1) 3 L sen3t ds
5 ds s 2 4 0

18 d 2 3
LFt 2
3 2
ds
5 ds s 4 s s 9
18 0 s 2 4 2 2s 3
LFt 3 ds
5 s2 4 s s 2
9
18 0 s2 4 2 2s 1 1 s
LFt 2
9 tan
5 s2 4 3 3 s

18 4s 1 1 s
LFt 2
3 tan tan
5 s2 4 3 3

72s 1 s
LFt 2
3 tan
5 s2 4 2 3
72s 1 s
LFt 2
3 3 tan
5 s 8s 16 2 3

3 3 5
c) F t L F" t si F t cos2t 2e 3t
t
4 5
3 3 5
Ft cos2t 2e 3t t
4 5
3 3 5 3 5
F0 cos 2 0 2e 3 0 0 2
4 5 4 4
5
F0
4
3 3 5
F't 2 sen 2t 2( 3) e 3t 5t
4 5
3
F't sen 2t 6 e 3t 3t 5
2
3 5
F'0 sen 2(0) 6 e 3( 0 ) 3 0
2
F'0 6

L F '' t S 2L F t S Y ( 0) Y ' ( 0 )
3 3 5 3t
LF t L cos 2t 2e
t
4 5
3 3 5
LF t L cos 2t 2 L e 3t Lt
4 5
3 S 1 3 5!
LF t 2
2
4 S 4 S 3 5 S6
3S 2 72
LF t
4 S2 4 S 3 S6
3S 2 72 5
L F '' t S2 S 6
4 S2 4 S 3 S6 4
3S 3 2S 2 72 5S
L F '' t 6
4 S2 4 S 3 S4 4

1
3.-Aplicar Tabla, simplificación y método correspondiente para determinar L f s F t

3
7 s 5
1 4 5s 5 7 7s 4 4 5
A) L 2 3
3 9 s 2 10s 25 s2 
8 18
  4
3 s 12  s2
7
c)
4
 
 
b)  
d)
a)

3 3
7 s 5 7 s
4 1 1 4 1 1 5
a) L 1 2
L 2
L 2
3 3 3 3 3
3 s 12 s 4 s 4
4 4 4

3
s
7 1 4 5 1 2
L 2
L 2
3 3 6 3
s 4 s 4
4 4
3 3
7 4 5 4t
e cosh 2t e senh 2t
3 6

5s 5 7 5s 5 7
b) L 1 3
L1
2 3
9 s2 10s 25 9 s 5
5s 5 7
L1 6 6
9 s 5 9 s 5
5 1 1 7 1 1
L 6
L 6
9 s 5 9 s 5
5 5t t 4 7 5t t 5 1 7
e e e 5t t 4 e 5t t 5
9 5! 9 6! 216 6480

7s 4 7s 4 7 1 s 1 1 1
c) L 1 L1 L L
8s 2 18 9 8 s2
9 2 s2
9
8 s2
4 4 4
3
7 1 s 1 2 1 2
L L
8 9 2 3 9
s2 s2
4 4
7 3 1 3
cosh t senh t
8 2 3 2
2
4 5 7 1 7
d )L 1 4 5 L 2
4 2 2
s2 s 2
7 7
4 35 2
sen t
2 7
2
2 35sen t
7

1 4s 7 6s 4
B) L
5 17 1
s2 s s2 s 20
3 4 3
2 2
5 17 5 5 5 17
s2 s s2 s
3 4 3 6 6 4
2
5 32
s
6 9
5 5
4 s 7
4s 7 6 6
2 2
5 32 5 32
s s
6 9 6 9
5 31 5
4 s 4 s
6 3 6
2 2 2
5 32 5 32
s s
6 9 6 3
5 32
4 s
6 31 3 3
2 2
5
2
32 3 32 5
2
32
2
s s
6 3 6 3

5 5
1 4s 7 6
t 32 31 6
t 32
L 4e c os t e sen t
5 17 3 32 3
s2 s
3 4

1 6s 4
L ?
2 1
s s 20
3
2 2
1 1 1
s2 s 20
3 6 6
2
1 719
s
6 36
1 1 1
6 s 4 6 s
6s 4 6 6 6 3
2 2 2 2
1 719 1 719 1 719 1 719
s s s s
6 36 6 36 6 36 6 36

1 719
6 s
6 6 6
2
3 2
2 2
1 719 719 1 719
s s
6 6 6 6

1
1 6s 4 6
t 719 18 719
L 6e cos t sen t
1 6 719 6
s2 s 20
3

1 s 2 2s 3
C)L
s2 2s 2 s 2 2s 5
s 2 25 3 As B Cs D
2
s 2s 2 s 2 2s 5 s2
2s 2 s 2
2s 5
s2 2s 3 As B s 2 2s 5 Cs D s2 2s 2
s2 2s 3 As3 2 As 2 5 As Bs 2 2 Bs 5 B Cs 3 2Cs 2 2Cs Ds 2 2 Ds 2 D
s2 2s 3 A B s 3 2A B 2C D s2 5 A 2B 2C 2 D s 5B 2D
A B 0
2A B 2C D 1
5A 2 B 2C D 2
5B 2D 3

2 A B 2C D 1 1
5 A 2 B 2C D 2 1

2 A B 2C D 1
5 A 2 B 2C D 2
3A B 1

A B 0 1
3A B 1
A B 0
3A B 1
1
2 A 1; A
2
1
B A; B
2
5B 2D 3
1
3 5
3 5B 2 11
D D
2 2 4
2 A B 2C D 1
1 1 11 17
2 2C 1; 2C 1
2 2 4 4
17 13 13
2C 1 ; 2C C
4 4 8

1 1 1 11
s s
L1 2 2 2 4
s 2 2s 2 s 2 2s 5

1 1 1 1 1 3
s s 1 1 s 1
2 2 2 2 2 2
2 2 2
s 1 1 s 1 1 s 1 1
1 1
s
2 2 1 1 s 1 3 1 1
L1 2
L 2
L 2
s 2s 2 2 s 1 1 2 s 1 1

1 t 3 t
e cost e sent
2 2
1 11 1 11
s s
L1 22 4 L1 2
2
4
s 2s 5 s 1 4

1 11 1 13
s 1 1 s 1
L1 2 4 L1 2 4
2 2
s 1 4 s 1 4

1 1 s 1 13 1 1 2
L 2
L 2
2 s 1 4 4 2 s 1 4
1 t 13 t
e cos 2t e sen 2t
2 8

2 5
4.- Utilizar el teorema de Convolución y determine: L1
s3 s 2 2
2 5
L1
s3 s2 2
1 1
2 5L 1
s3 s2 2
1 1 t2
L
s3 2
1 1 2
L1 L1
s2 2 2 s2 2
1 1
L1 sen 2t
s2 2 2
2
1 2 5 t
L 2 5 sen 2 t d
s3 s2 2 0 2

2 5 t
L1 5 2
sen 2 t d
0
s3 s2 2
2
u dv sen 2 t
1
du 2 v cos 2 t
2
2 5 5
L1 2
cos 2 t 2 cos 2t d
3 2
s s 2 2

2 1
1 2 5 5 cos 2 t 2 sen t 2 sen 2
L 2
s3 s2 2 2

2 5 5 1
L1 t 2 cos 2t 2t sen (t ) 2 sen 2 t
3 2
s s 2 2 2

5.- Determine el semiperiodo del seno de Fourier para F x 4x ; 0 x 1 realizar el espectro
de la función.

1 t 1 1
bn f ( x)dx 4 x sen(nx) dx
T 0 1 0
u 4x dv sen nx dx
cos nx
du 4dx v
n
cos nx cos nx
bn 4x
n n
1
cos nx 4
bn 4x sen (nx)
n n2 0

cos n 4
bn 4 sen (n)
n n

6.-DESARROLLE LA EXPANSIÓN Y REALICE EL ESPECTRO DE FOURIER DE LA FUNCIÓN
1 si 0 x 1
F x T 2
2 x si 1 x 2
1 1 2
A0 dx 2 x dx
2 0 1

2
1 1 x2
A0 x 2x
2 0 2 1

1 12 12
A0 1 0 2.2 2. 1
2 2 2
1 1 1
A0 1 4 2
2 2 2
1
A0 3
2
3
A0
2
1 1 cos nx 2 cos nx
An dx 2 x dx
2 0 2 1 2
1 2
1 2 sen nx 2(2 x) sen nx cos nx
An
2 n 2 0 n 2 2 1

1 2 3
An
n n n
1 1 2
bn sen nxdx 2 x sen nxdx
2 0 1

1
1 cos nx cos nx sennx
bn 2 x
2 n 2 2 0
1
bn n
2

Asig de transf_y_fourier avanzado

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