Page 139 - Schaum's Outline of Differential Equations

122 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS [CHAP. 14

This problem is, therefore, an example of forced undamped motion. The solution to the associated homogeneous
equation is
x h = GI cos 4t + c 2 sin 4t

A particular solution is found by the method of undetermined coefficients (the modification described in Chapter 11
is necessary here): x =-\ cos 4t. The solution to (1) is then

Applying the initial conditions, x(0) = - y and i(0) = 0, we obtain

Note that \x\ —> °° as t —> °°. This phenomenon is called pure resonance. It is due to the forcing function F(t)
having the same circular frequency as that of the associated free undamped system.

14.11. Write the steady-state motion found in Problem 14.9 in the form specified by Eq. (14.13).
The steady-state displacement is given by (2) of Problem 14.9 as

Its circular frequency is to = 1. Here

and

The coefficient of the cosine term in the steady-state displacement is negative, so k= 1, and Eq. (14.13) becomes

14.12. An RCL circuit connected in series has R = 180 ohms, C = 1/280 farad, L = 20 henries, and an applied
voltage E(i) = 10 sin t. Assuming no initial charge on the capacitor, but an initial current of 1 ampere at
t = 0 when the voltage is first applied, find the subsequent charge on the capacitor.
Substituting the given quantities into Eq. (14.5), we obtain

This equation is identical in form to (1) of Problem 14.9; hence, the solution must be identical in form to the
solution of that equation. Thus,

Applying the initial conditions q(0) = 0 and q(0) = 1, we obtain Cj = 110/500 and c 2 = -101/500. Hence,

As in Problem 14.9, the solution is the sum of transient and steady-state terms.

2
14.13. An RCL circuit connected in series has R = 10 ohms, C = 10 farad, L = jhenry, and an applied voltage
E = 12 volts. Assuming no initial current and no initial charge at t = 0 when the voltage is first applied,
find the subsequent current in the system.