Page 81 - Schaum's Outline of Differential Equations

64 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 7

7.16. A tank initially holds 100 gal of a brine solution containing 20 Ib of salt. At t = 0, fresh water is poured
into the tank at the rate of 5 gal/min, while the well-stirred mixture leaves the tank at the same rate. Find
the amount of salt in the tank at any time t.
Here, V 0 = 100, a = 20, b = 0, and e =/= 5. Equation (7.8) becomes

The solution of this linear equation is

At t=0, we are given that Q = a = 20. Substituting these values into (1), we find that c = 20, so that (1) can be
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rewritten as Q = 20e~' . Note that as t —* °°, Q —* 0 as it should, since only fresh water is being added.

7.17. A tank initially holds 100 gal of a brine solution containing 1 Ib of salt. At t = 0 another brine solution
containing 1 Ib of salt per gallon is poured into the tank at the rate of 3 gal/min, while the well-stirred
mixture leaves the tank at the same rate. Find (a) the amount of salt in the tank at any time t and (b) the
time at which the mixture in the tank contains 2 Ib of salt.
(a) Here V Q = 100, a = 1, b = 1, and e =/= 3; hence, (7.8) becomes

The solution to this linear differential equation is

At t = 0, Q = a=l. Substituting these values into (1), we find 1 = ce° + 100, or c = -99. Then (1) can be
rewritten as

(b) We require t when 2 = 2. Substituting 2 = 2 into (2), we obtain

from which

7.18. A 50-gal tank initially contains 10 gal of fresh water. At t = 0, a brine solution containing 1 Ib of salt per
gallon is poured into the tank at the rate of 4 gal/min, while the well-stirred mixture leaves the tank at
the rate of 2 gal/min. Find (a) the amount of time required for overflow to occur and (b) the amount of
salt in the tank at the moment of overflow.
(a) Here a = 0, b = 1, e = 4,f= 2, and V Q = 10. The volume of brine in the tank at any time t is given by (7.7) as
V 0 + et -ft = 10 + 2t. We require t when 10 + 2t = 50; hence, t = 20 min.
(b) For this problem, (7.8) becomes

This is a linear equation; its solution is given in Problem 6.13 as

At t = 0, Q = a = 0. Substituting these values into (1), we find that c = 0. We require Q at the moment of