Page 71 - Schaum's Outline of Differential Equations

54 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 7

At t= 0, N(0) = 20,000, which when substituted into (1) yields

With this value of c, (1) becomes

Equation (2) gives the dollar balance in the account at any time t.
(a) Substituting t = 3 into (2), we find the balance after three years to be

(b) We seek the time t at which N(t) = $40,000. Substituting these values into (2) and solving for t, we obtain

7.2. A person places $5000 in an account that accrues interest compounded continuously. Assuming no
additional deposits or withdrawals, how much will be in the account after seven years if the interest
rate is a constant 8.5 percent for the first four years and a constant 9.25 percent for the last three years?
Let N(t) denote the balance in the account at any time t. Initially, N(0) = 5000. For the first four years, k = 0.085
and Eq. (7.1) becomes

Its solution is

At t= 0, N(0) = 5000, which when substituted into (1) yields

and (1) becomes

Substituting t = 4 into (2), we find the balance after four years to be

This amount also represents the beginning balance for the last three-year period.
Over the last three years, the interest rate is 9.25 percent and (7.1) becomes

Its solution is

Ait = 4, N(4) = $7024.74, which when substituted into (3) yields

and (3) becomes

Substituting t = 7 into (4), we find the balance after seven years to be