Page 142 - Schaum's Outline of Differential Equations

CHAP. 14] SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 125

whereupon

and c 2 = -0.0324. Substituting this value into (2) and simplifying, we obtain as before

14.17. Determine the circular frequency, the natural frequency, and the period of the steady-state current found
in Problem 14.16.
The current is given by (3) of Problem 14.16. As t —> °°, the exponential terms tend to zero, so the steady-state
current is

Circular frequency:

Natural frequency:
Period:

14.18. Write the steady-state current found in Problem 14.17 in the form specified by Eq. (14.13).
The amplitude is

and the phase angle is

The circular frequency is to= 100. The coefficient of the cosine term is positive, so k= 0 and Eq. (14.13) becomes

14.19. Determine whether a cylinder of radius 4 in, height 10 in, and weight 15 Ib can float in a deep pool of
3
water of weight density 62.5 Ib/ft .
Let h denote the length (in feet) of the submerged portion of the cylinder at equilibrium. With r = jft, it follows
from Eq. (14.9) that

Thus, the cylinder will float with 10 - 8.25 = 1.75 in of length above the water line at equilibrium.

14.20. Determine an expression for the motion of the cylinder described in Problem 14.19 if it is released with
20 percent of its length above the water line with a velocity of 5 ft/sec in the downward direction.
3
Here r = }ft, p = 62.5 Ib/ft , m = 15/32 slugs and Eq. (14.10) becomes