Assignment 1 Solution

Assignment # 1 (solution) Exercise 4.1

13.

L {f (t)} =

∫

∞

0

∞ ⎛ 1 ⎞ 1 te 4t e − st dt = ∫ te ( 4− s )t dt = ⎜⎜ te ( 4− s )t − e ( 4− s )t ⎟⎟ 2 0 (4 − s) ⎝4−s ⎠

25. L {t 3 + 3t 2 + 3t + 1} = 3! + 3 2 + 3 + 1 4 3 2

s

s

s

s

31. L {4t 2 − 5sin 3t} = 4 2 − 5 3 s3 s2 + 9 Exercise 4.2

5. L

(s + 1) 3 ⎫⎪ L ⎬= ⎨ ⎪⎩ s 4 ⎪⎭

-1 ⎧ ⎪

-1 ⎧ 1

⎨ ⎩s

+3

1 3 2 1 3! ⎫ 3 1 + + 4 ⎬ = 1 + 3t + t 2 + t 3 2 3 s 2s 6s ⎭ 2 6

1

∞ 0

=

1 ,s > 4 (4 − s ) 2

11. L

-1

⎧ ⎨ 2 ⎩s

5 ⎫ L ⎬= + 49 ⎭

-1 ⎧ 5

⎨ ⎩7

7 ⎫ 5 ⎬ = sin 7t s + 49 ⎭ 7 2

Exercise 4.3

7. L {e t sin 3t} =

3 (s −1) 2 + 9

⎫ ⎧ 19. L--1 ⎧⎪ 2s − 1 ⎫⎪ = L—1 ⎪⎨ 5 − 1 − 5 − 4 − 3 2 ⎪⎬ = 5 − t − 5e−t − 4te−t − 3 t 2e−t ⎨ 2 ⎬ 2 3 s + 1 (s + 1) 2 2 (s + 1) 3 ⎪⎭ 2 ⎪⎩ s s ⎪⎩ s ( s + 1) ⎪⎭

2

39. L- {tU (t − 2)} = L {(t − 2)U (t − 2) + 2U (t − 2)} = e

−2 s

s2

--1

43. L

e −2 s { 3 } = L—1 {1 23 e −2 s } = 1 (t − 2) 2U (t − 2) s 2s 2

Exercise 4.4

7

21. L- {e − t ∗ et cos t} =

s −1 (s + 1)[(s − 1) 2 + 1]

25.

27. L- {∫0 τ e t −τ dτ } = L- {t} L-{e t } = t

32.

3

1 s (s − 1) 2

+

2e −2 s s