1550078481-Ordinary_Differential_Equations__Roberts_

284 Ordinary Differential Equations

motion the solution (19) may be rewritten in the equivalent form

(20) y(t) = Ae-at sin (wt+¢)

where A and¢ are arbitrary constants. The factor Ae-at is called the damp-

ing factor. The factor sin (wt + ¢) represents periodic, oscillatory motion
with amplitude l. Since a> 0 and I sin (wt+ ¢)1~1, y(t)---+ 0 as t---+ oo. So
the product of the two factors Ae-at and sin (wt+¢) represents oscillatory
motion in which the amplitude of oscillation decreases with increasing time.
The time interval between two successive maxima is still called the period, P,

and as in the free undamped case P = 2-rr/w. See Figure 6.5.

I-Period-I

y =Ae-at

t

y=-Ae-at

y = Ae -a t sin ( ro t + )

Figure 6.5 Damped, Oscillatory Motion

Case 2. When b^2 - 4ad = 0, the system is said to be critically damped.

In this case, the roots of the auxiliary equation are real and equal, r 1 = r 2 =

-b/(2a). So the general solution of (12) is

(21) y = (A+ Bt)e-at

where a = b/(2a) > 0 and A and B are arbitrary constants. Due to the

damping factor e-at and the fact that a > 0, the solution y(t) ---+ 0 as t ---+ oo.

Since the factor A+ Bt is linear, the motion is not oscillatory and, in fact, the

solution can cross the t-axis (the time axis) at most once. The graph of (21)
depends on the constants A and B which, of course, are determined by the
initial conditions. Three typical graphs of equation (21)- critically damped
motion- are sketched in Figure 6.6.

Case 3. When b^2 - 4ad > 0, the system is said to be overdamped. The

roots of the auxiliary equation are real, distinct, and negative. See equa-
tion (13). So the general solution of (12) is

(22)