Page 121 - Advanced engineering mathematics

3.4 Convolution 101

25

20

15

10

5

0
0 2 4 6 8 10
t
FIGURE 3.21 Replacement function.

Figure 3.21 is a graph of this replacement function for A = 2, B = 0.001, and k = 1. As
expected, r(t) is a strictly increasing function, because it measures total replacements up to a
given time. The graph gives an indication of how the drug needs to be replenished to maintain
f (t) doses at time t.

SECTION 3.4 PROBLEMS

In each of Problems 1 through 8, use the convolution the- In each of Problems 9 through 16, use the convolution
orem to help compute the inverse Laplace transform of theorem to write a formula for the solution in terms of f .
the function. Wherever they occur, a and b are positive

constants. 9. y − 5y + 6y = f (t); y(0) = y (0) = 0
1 10. y + 10y + 24y = f (t); y(0) = 1, y (0) = 0

1.
(s + 4)(s − 4) 11. y − 8y + 12y = f (t); y(0) =−3, y (0) = 2
2
2

1

2. e −2s 12. y − 4y − 5y = f (t); y(0) = 2, y (0) = 1

2
s + 16

s 13. y + 9y = f (t); y(0) =−1, y (0) = 1
3.
2
(s + a )(s + b ) 14. y − k y = f (t); y(0) = 2, y (0) =−4
2
2
2
2

s 2 15. y (3) − y − 4y + 4y = f (t); y(0) = y (0) =

4.
2
(s − 3)(s + 5) 1, y (0) = 0

1 (4)
5. 16. y − 11y + 18y = f (t); y(0) = y (0) = y (0) =
2
2 2
(3)
s(s + a ) y (0) = 0
1
6.
s (s − 5) In each of Problems 17 through 23, solve the integral
4
equation.
1
7. e −4s
s(s + 2)
t
17. f (t) =−1 + f (t − τ)e −3τ dτ
2 0
8.
t f (t − τ)sin(τ)dτ
2
2
s (s + 5) 18. f (t) =−t + 0
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