FIGURE N9–10
Figure N9–10 shows the slope field for equation (3), with k = 0.00179, which was obtained by
solving equation (5) above. Note that the slopes are the same along any horizontal line, and that
they are close to zero initially, reach a maximum at P = 500, then diminish again as P approaches
its limiting value, 1000. We have superimposed the solution curve for P(t) that we obtained in
(6) above.BC ONLY
EXAMPLE 21
Suppose a flu-like virus is spreading through a population of 50,000 at a rate proportional both
to the number of people already infected and to the number still uninfected. If 100 people were
infected yesterday and 130 are infected today:
(a) write an expression for the number of people N(t) infected after t days;
(b) determine how many will be infected a week from today;
(c) indicate when the virus will be speading the fastest.
SOLUTIONS:
(a) We are told that N ′(t) = k · N · (50,000 −N), that N(0) = 100, and that N(1) = 130. The d.e.
describing logistic growth leads to
From N(0) = 100, we get
which yields c = 499. From N(1) = 130, we get