Page 72 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 55

7.3. What constant interest rate is required if an initial deposit placed into an account that accrues interest
compounded continuously is to double its value in six years?
The balance N(t) in the account at any time t is governed by (7.1)

which has as its solution

We are not given an amount for the initial deposit, so we denote it as N 0. At t = 0, N(0) = N 0, which when
substituted into (1) yields

and (1) becomes

We seek the value of k for which N=2N 0 when t = 6. Substituting these values into (2) and solving for k,
we find

An interest rate of 11.55 percent is required.

7.4. A bacteria culture is known to grow at a rate proportional to the amount present. After one hour, 1000
strands of the bacteria are observed in the culture; and after four hours, 3000 strands. Find (a) an expres-
sion for the approximate number of strands of the bacteria present in the culture at any time t and (b) the
approximate number of strands of the bacteria originally in the culture.
(a) Let N(t) denote the number of bacteria strands in the culture at time t. From (6.1), dNIdt — kN = 0, which is
both linear and separable. Its solution is

AU=1,N= 1000; hence,

At t = 4, W = 3000; hence,

Solving (2) and (3) for k and c, we find

Substituting these values of k and c into (1), we obtain

as an expression for the amount of the bacteria present at any time t.
(b) We require N at t = 0. Substituting t = 0 into (4), we obtain N(0) = 694e (0 366)(0) = 694.