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

1168 chapter 17 Second-Order Differential Equations
19–22 Solve the differential equation using (a) undetermined
coefficients and (b) variation of parameters.
19. 4y0 1 y − cos x 20. y0 2 2y9 2 3y − x 1 2
21. y0 2 2y9 1 y − e 2x
22. y0 2 y9 − e x
23–28 Solve the differential equation using the method of variation
of parameters.
23. y0 1 y − sec 2 x, 0 , x , y2
24. y0 1 y − sec 3 x, 0 , x , y2
25. y0 2 3y9 1 2y −
1
1 1 e 2x
26. y0 1 3y9 1 2y − sinse x d
27. y0 2 2y9 1 y −
e x
1 1 x 2
28. y0 1 4y9 1 4y − e22x
x 3
Second-order linear differential equations have a variety of applications in science and
engineering. In this section we explore two of them: the vibration of springs and electric
circuits.
Vibrating Springs
We consider the motion of an object with mass m at the end of a spring that is either vertical
(as in Figure 1) or horizontal on a level surface (as in Figure 2).
In Section 6.4 we discussed Hooke’s Law, which says that if the spring is stretched (or
compressed) x units from its natural length, then it exerts a force that is proportional to x:
m
equilibrium
position
0
restoring force − 2kx
FIGURE 1
x
x
m
where k is a positive constant (called the spring constant). If we ignore any external
resisting forces (due to air resistance or friction) then, by Newton’s Second Law (force
equals mass times acceleration), we have
1 m d 2 x
dt 2 − 2kx or m d 2 x
dt 2 1 kx − 0
equilibrium position
This is a second-order linear differential equation. Its auxiliary equation is mr 2 1 k − 0
with roots r − 6i, where − skym . Thus the general solution is
FIGURE 2
m
0 x
x
which can also be written as
xstd − c 1 cos t 1 c 2 sin t
xstd − A cosst 1 d
where − skym (frequency)
A − sc 12 1 c 2
2
(amplitude)
cos − c 1
A
sin − 2 c 2
A
s is the phase angled
(See Exercise 17.) This type of motion is called simple harmonic motion.
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