Page 82 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 65

overflow, which from part (a) is t = 20. Thus,

7.19. An RL circuit has an emf of 5 volts, a resistance of 50 ohms, an inductance of 1 henry, and no initial
current. Find the current in the circuit at any time t.
Here E = 5, R = 50, and L = 1; hence (7.9) becomes

This equation is linear; its solution is

thus . The current at any time t is then

50
The quantity —-^e ' in (_/) is called the transient current, since this quantity goes to zero ("dies out") as t —> °°.
The quantity -^ in (_/) is called the steady-state current. As t —> °°, the current I approaches the value of the steady-
state current.

7.20. An RL circuit has an emf given (in volts) by 3 sin 2t, a resistance of 10 ohms, an inductance of 0.5 henry,
and an initial current of 6 amperes. Find the current in the circuit at any time t.
Here, E = 3 sin 2t, R = 10, and L = 0.5; hence (7.9) becomes

This equation is linear, with solution (see Chapter 6)

Carrying out the integrations (the second integral requires two integrations by parts), we obtain

At t = 0, 1 = 6; hence,

whence c = 609/101. The current at any time t is

As in Problem 7.18, the current is the sum of a transient current, here (609/101)e 20( , and a
steady-state current,