Page 69 - Schaum's Outline of Differential Equations

52 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 7

(See Problem 7.11.) When k > 0, the limiting velocity V; is defined by

Caution: Equations (7.4), (7.5), and (7.6), are valid only if the given conditions are satisfied. These equa-
tions are not valid if, for example, air resistance is not proportional to velocity but to the velocity squared, or if
the upward direction is taken to be the positive direction. (See Problems 7.14 and 7.15.)

DILUTION PROBLEMS

Consider a tank which initially holds V 0 gal of brine that contains a Ib of salt. Another brine solution,
containing b Ib of salt per gallon, is poured into the tank at the rate of e gal/min while, simultaneously, the
well-stirred solution leaves the tank at the rate of/gal/min (Fig. 7-2). The problem is to find the amount of salt
in the tank at any time t.
Let <2 denote the amount (in pounds) of salt in the tank at any time t. The time rate of change of Q, dQIdt,
equals the rate at which salt enters the tank minus the rate at which salt leaves the tank. Salt enters the tank at
the rate of be Ib/min. To determine the rate at which salt leaves the tank, we first calculate the volume of brine
in the tank at any time t, which is the initial volume V 0 plus the volume of brine added et minus the volume of
brine removed t. Thus, the volume of brine at any time is
f

The concentration of salt in the tank at any time is QI(V 0 + et -ft), from which it follows that salt leaves the
tank at the rate of

Thus,

or

(See Problems 7.16-7.18.)

ELECTRICAL CIRCUITS
The basic equation governing the amount of current I (in amperes) in a simple KL circuit (Fig. 7-3)
consisting of a resistance R (in ohms), an inductor L (in henries), and an electromotive force (abbreviated emf)
E (in volts) is

For an RC circuit consisting of a resistance, a capacitance C (in farads), an emf, and no inductance (Fig. 7-4),
the equation governing the amount of electrical charge q (in coulombs) on the capacitor is

The relationship between q and I is

(See Problems 7.19-7.22.) For more complex circuits see Chapter 14.