Page 134 - Schaum's Outline of Differential Equations

CHAP. 14] SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 117

Fig. 14.3

Motion will occur when the cylinder is displaced from its equilibrium position. We arbitrarily take the
upward direction to be the positive ^-direction. If the cylinder is raised out of the water by x(t) units, as shown
in Fig. 14-3, then it is no longer in equilibrium. The downward or negative force on such a body remains mg
2
but the buoyant or positive force is reduced to Jtr [h - x(t)]p. It now follows from Newton's second law that

Substituting (14.9) into this last equation, we can simplify it to

or

(See Problems 14.19-14.24.)

CLASSIFYING SOLUTIONS

Vibrating springs, simple electrical circuits, and floating bodies are all governed by second-order linear
differential equations with constant coefficients of the form

For vibrating spring problems defined by Eq. (14.1), a 1 = aim, a 0 = him, and/(?) = F(t)lm. For buoyancy problems
2
defined by Eq. (14.10), a 1 = 0, a 0 = Jtr plm, and/(?) = 0. For electrical circuit problems, the independent variable
x is replaced either by q in Eq. (14.5) or I in Eq. (14.7).
The motion or current in all of these systems is classified as free and undamped when/(?) = 0 and a 1 = 0.
It is classified as free and damped when/(?) is identically zero but a 1 is not zero. For damped motion, there are
three separate cases to consider, depending on whether the roots of the associated characteristic equation (see
Chapter 9) are (1) real and distinct, (2) equal, or (3) complex conjugate. These cases are respectively classified
as (1) overdamped, (2) critically damped, and (3) oscillatory damped (or, in electrical problems, underdamped).
If/(O is not identically zero, the motion or current is classified as forced.
A motion or current is transient if it "dies out" (that is, goes to zero) as t —> °°. A steady-state motion or
current is one that is not transient and does not become unbounded. Free damped systems always yield transient