Page 140 - Schaum's Outline of Differential Equations

CHAP. 14] SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 123

Substituting the given values into Eq. (14.7), we obtain the homogeneous equation [since E(t) = 12, dEldt = 0]

The roots of the associated characteristic equation are A,j = -10 + 10; and X 2 = -10 - 10;; hence, this is an example
of a free underdamped system for the current. The solution is

The initial conditions are 7(0) = 0 and, from Eq. (14.8),

10
Applying these conditions to (7), we obtain c l = 0 and c 2 =y; thus, I = ^e ' sinlOf, which is completely
transient.

14.14. Solve Problem 14.13 by first finding the charge on the capacitor.
We first solve for the charge q and then use 7 = dqldt to obtain the current. Substituting the values given in
Problem 14.13 into Eq. (14.5), we have q + 20q + 200q = 24, which represents a forced system for the charge, in
contrast to the free damped system obtained in Problem 14.3 for the current. Using the method of undetermined
coefficients to find a particular solution, we obtain the general solution

Initial conditions for the charge are q(0) = 0 and q(0) = 0; applying them, we obtain Cj = c 2 = -3/25. Therefore,

and

as before.
Note that although the current is completely transient, the charge on the capacitor is the sum of both transient
and steady-state terms.

14.15. An RCL circuit connected in series has a resistance of 5 ohms, an inductance of 0.05 henry, a capacitor
4
of 4 X 10~ farad, and an applied alternating emf of 200 cos 100? volts. Find an expression for the current
flowing through this circuit if the initial current and the initial charge on the capacitor are both zero.
Here RIL = 5/0.05 = 100, 1/(LC) = 1/[0.05(4 X 10^)] = 50,000, and

so Eq. (14.7) becomes

The roots of its characteristic equation are -50 + 50Vl9i, hence the solution to the associated homogeneous
problem is

Using the method of undetermined coefficients, we find a particular solution to be