Page 133 - Schaum's Outline of Differential Equations

116 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONSIONS {CHAP.14

connected in series. The current 7 flowing through the circuit is measured in amperes and the charge q on the
capacitor is measured in coulombs.
Kirchhojfs loop law: The algebraic sum of the voltage drops in a simple closed electric circuit is zero.
It is known that the voltage drops across a resistor, a capacitor, and an inductor are respectively RI, (HC)q,
and L(dlldt) where q is the charge on the capacitor. The voltage drop across an emf is —E(t). Thus, from
Kirchhoff s loop law, we have

The relationship between q and 7 is

Substituting these values into (14.3), we obtain

The initial conditions for q are

To obtain a differential equation for the current, we differentiate Eq. (14.3) with respect to t and then
substitute Eq. (14.4) directly into the resulting equation to obtain

The first initial condition is 7(0) = 7 0. The second initial condition is obtained from Eq. (14.3) by solving for
dlldt and then setting t = 0. Thus,

An expression for the current can be gotten either by solving Eq. (14.7) directly or by solving Eq. (14.5) for
the charge and then differentiating that expression. (See Problems 14.12-14.16.)

BUOYANCY PROBLEMS

Consider a body of mass m submerged either partially or totally in a liquid of weight density p. Such a body
experiences two forces, a downward force due to gravity and a counter force governed by:

Archimedes' principle: A body in liquid experiences a buoyant upward force equal to the weight of the liquid
displaced by that body.
Equilibrium occurs when the buoyant force of the displaced liquid equals the force of gravity on the body.
Figure 14-3 depicts the situation for a cylinder of radius r and height 77 where h units of cylinder height are
2
submerged at equilibrium. At equilibrium, the volume of water displaced by the cylinder is 7tr h, which provides
2
a buoyant force of 7tr hp that must equal the weight of the cylinder mg. Thus,