Page 74 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 57

(c) We require t when N = 50/2 = 25. Substituting N=25 into (3) and solving for t, we find

The time required to reduce a decaying material to one half its original mass is called the half-life of the
material. For this problem, the half-life is 13 hours.

7.7. Five mice in a stable population of 500 are intentionally infected with a contagious disease to test
a theory of epidemic spread that postulates the rate of change in the infected population is proportional
to the product of the number of mice who have the disease with the number that are disease free.
Assuming the theory is correct, how long will it take half the population to contract the disease?
Let N(t) denote the number of mice with the disease at time t. We are given that N(0) = 5, and it follows that
500 - N(t) is the number of mice without the disease at time t. The theory predicts that

where k is a constant of proportionality. This equation is different from (7.1) because the rate of change is no longer
proportional to just the number of mice who have the disease. Equation (1) has the differential form

which is separable. Using partial fraction decomposition, we have

hence (2) may be rewritten as

Its solution is

or

which may be rewritten as

Bute Setting we can write (3) as

At t= 0, N= 5. Substituting these values into (4), we find

so GI = 1/99 and (4) becomes