Lista de Exercı́cios
Transformada de Laplace
Nos exercı́cios 1 a 7, usando a definição de Transformada de Laplace, mostre que:
1
1. L{1} = , s > 0
s
1
2. L{eat } =
,s>a
s−a
n
3. L{t } =
n!
sn+1
5. L{cos(at)} =
s
,s>0
s2 + a2
6. L{ senh (at)} =
a
s2 − a2
,s>a
, s > 0, n ∈ N
4. L{ sen (at)} =
a
s 2 + a2
√
Z +∞
8. Sabendo que
7. L{cosh(at)} =
,s>0
e
−x2
0
π
dx =
, mostre que L{t−1/2 } =
2
r
s
s 2 − a2
,s>a
π
, s > 0.
s
Nos exercı́cios 9 a 14, calcule as transformadas de Laplace.
9. L{e3t cos(2t)}
12. L{t2 sen (t)}
10. L{cos2 (at)}, a ∈ R
13. L{t3 e−t }
11. L{ sen (5t) cos(2t)}
14. L{teat cos(bt)}, a, b ∈ R
Nos exercı́cios 15 a 17, usando propriedades das transformadas de Laplace, calcule:
100
15. L{et sen (3t)}
16. L{t5 e4t }
t d
−t 100
17. L e 100 e t
dt
18. Determine a transformada de Laplace L{t5/2 }.
Mostre que L
Laplace.
sen (t)
19. L
t
n
f (t)
t
o
=
R +∞
s
F (u) du e nos exercı́cios 19 a 21, calcule as transformadas de
20. L
cos(at) − 1
t
,a ∈ R
at
e − e−at
21. L
,a ∈ R
t
Nos exercı́cios 22 a 27, calcule as transformadas de Laplace inversa.
−πs e
22. L−1 (s − 2)−2
−1
25. L
2
s + 16
7
7
4
23. L−1
+
−1
3
2
26.
L
arctan
(s − 1)
(s + 1) − 4
s
s
1
−1
−1
24. L
27. L
(s + 1)2 (s2 + 1)
((s2 + 1)2
1
Nos exercı́cios 28 a 39, use a transformada de Laplace para resolver os seguintes problemas de
valor inicial em [0, +∞)
00
00
y + 4y 0 + 4y = e−t
y − 9y 0 = 5e−2t
28.
34.
0
y(0) = 0, y (0) = 1
y(0) = 1, y 0 (0) = 2
00
00
y + 4y 0 + 3y = 0
y − 5y 0 + 6y = 3e3t
29.
35.
0
y(0) = 0, y (0) = 1
y(0) = 0, y 0 (0) = 0
00
00
y + 6y 0 − 7y = 0
y + 4y = 9t
30.
36.
0
y(0) = 1, y (0) = 0
y(0) = 0, y 0 (0) = 7
00
00
y − y 0 − 2y = t
y + y = cos t
31.
37.
y(0) = 0, y 0 (0) = 0
y(0) = 0, y 0 (0) = −1
iv
000
y − k4 y = 0
y − 4y 0 = senh t
32.
38.
0
00
000
y(0) = 1, y (0) = y (0) = y (0) = 0
y(0) = 0, y 0 (0) = 0, y 00 (0) = 0
00
00
y − 2y 0 + 5y = 0
y + y 0 − 2y = 5e−t sen (2t)
33.
39.
0
y(0) = 0, y (0) = 1
y(0) = 1, y 0 (0) = 0
2
RESPOSTAS DA LISTA de Transformada de Laplace
s−3
9. F (s) = (s−3)
2 +4
1
s
+ 2(s2 +4a)
10. F (s) = 2s
2
7
+ s23+9
11. F (s) = 12 s2 +49
2
24. f (t) = 21 ( sen (t) − te−t )
0
,0≤t<π
25. f (t) =
1
sen
(4t)
,
t≥π
4
26. f (t) = 1t sen (4t)
−2
12. F (s) = (s6s2 +1)
3
27. f (t) = 21 ( sen t − t cos t)
6
13. F (s) = (s+1)
4
28. y(t) = e−t − e−2t
(s−a)2 −b2
14. F (s) = [(s−a)
2 +b2 ]2
29. y(t) = 21 e−t − e−3t
√ 30. y(t) = 16 7 − cos( 6t)
15. F (s) = (s−1)3 2 +9
1 2t
31. y(t) = 41 − 12 t − 13 e−t + 12
e
5!
16. F (s) = (s−4)
6
32. y(t) = 41 ekt + 14 e−kt + 12 cos(kt)
100
17. F (s) = 100!(s−1)
s101
15 √
18. F (s) = 8
33. y(t) = 21 et sen (2t)
1
π s7/2
19. F (s) = π2 − arctan s, s > 0
√
20. F (s) = − ln
s2 +a2
s
,s>0
3 9t
5 −2t
34. y(t) = 21 + 11
e + 22
e
35. y(t) = 3te3t − 3e3t + 3e2t
36. y(t) = 49 t + 19
8 sen (2t)
21. F (s) = ln s+a
s−a , s > |a|
37. y(t) = − sen t + 21 t sen t
22. f (t) = te2t
1
38. y(t) = 14 − 13 cosh t 12
cosh(2t)
23. f (t) = 72 t2 et + 12 e−t sinh(2t)
13 t 1 −2t 1 −t
e − 3 e + 4 e cos(2t)− 43 e−t sen (2t)
39. y(t) = 12
3