James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1172 Chapter 17 Second-Order Differential Equations
A commonly occurring type of external force is a periodic force function
Fstd − F 0 cos 0 t where 0 ± − skym
In this case, and in the absence of a damping force (c − 0), you are asked in Exercise 9
to use the method of undetermined coefficients to show that
6 xstd − c 1 cos t 1 c 2 sin t 1
F 0
ms 2 2 0 2 d cos 0t
If 0 − , then the applied frequency reinforces the natural frequency and the result is
vibrations of large amplitude. This is the phenomenon of resonance (see Exercise 10).
Electric Circuits
E
switch
R
C
L
In Sections 9.3 and 9.5 we were able to use first-order separable and linear equations to
analyze electric circuits that contain a resistor and inductor (see Figure 9.3.5 or Figure
9.5.4) or a resistor and capacitor (see Exercise 9.5.29). Now that we know how
to solve second-order linear equations, we are in a position to analyze the circuit
shown in Figure 7. It contains an electromotive force E (supplied by a battery or generator),
a resistor R, an inductor L, and a capacitor C, in series. If the charge on the
capacitor at time t is Q − Qstd, then the current is the rate of change of Q with respect
to t: I − dQydt. As in Section 9.5, it is known from physics that the voltage drops across
the resistor, inductor, and capacitor are
FIGURE 7
RI
L dI
dt
Q
C
respectively. Kirchhoff’s voltage law says that the sum of these voltage drops is equal to
the supplied voltage:
L dI
dt 1 RI 1 Q C − Estd
Since I − dQydt, this equation becomes
7 L d 2 Q
dt 2
1 R dQ
dt
1 1 C Q − Estd
which is a second-order linear differential equation with constant coefficients. If the
charge Q 0 and the current I 0 are known at time 0, then we have the initial conditions
Qs0d − Q 0 Q9s0d − Is0d − I 0
and the initial-value problem can be solved by the methods of Section 17.2.
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