Page 83 - Schaum's Outline of Differential Equations

66 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 7

7.21. Rewrite the steady-state current of Problem 7.20 in the form A sin (2t - (f>). The angle (f> is called the
phase angle.
Since A sin (2t - (f>) = A(sm 2t cos (f> - cos 2t sin (f>), we require

Thus, A cos d> = and A sin d> = . It now follows that

and

has the required form if
Consequently, I s

7.22. An RC circuit has an emf given (in volts) by 400 cos 2t, a resistance of 100 ohms, and a capacitance of
2
10~ farad. Initially there is no charge on the capacitor. Find the current in the circuit at any time t.
We first find the charge q and then use (7.11) to obtain the current. Here, E = 4QOcos2t, R= 100, and
2
C= 10~ ; hence (7.10) becomes

This equation is linear, and its solution is (two integrations by parts are required)

At t = 0, q = 0; hence,

Thus

and using (7.11), we obtain

2
2
2
7.23. Find the orthogonal trajectories of the family of curves x + y = c .
2
2
The family, which is given by (7.12) with F(x, y, c) = x + y 2 — c , consists of circles with centers at the origin
and radii c. Implicitly differentiating the given equation with respect to x, we obtain