Second-Order Linear Differential Equations

1116 CHAPTER 15 Differential Equations
Vibrating Spring In Exercises 39–44, describe the motion of
a 32-pound weight suspended on a spring. Assume that the
2
weight stretches the spring foot from its natural position.
39. The weight is pulled 2 foot below the equilibrium position and
released.
2
40. The weight is raised 3 foot above the equilibrium position and
released.
2
41. The weight is raised 3 foot above the equilibrium position
1
and started off with a downward velocity of 2 foot per second.
1
42. The weight is pulled 2 foot below the equi-
1
librium position and started off with an upward velocity of 2
foot per second.
1
43. The weight is pulled 2 foot below the equilibrium position and
released. The motion takes place in a medium that furnishes a
1
damping force of magnitude 8 speed at all times.
1
44. The weight is pulled 2 foot
below the equilibrium position and released. The motion takes
place in a medium that furnishes a damping force of magnitude
1
at all times.
4 v
Vibrating Spring In Exercises 45–48, match the differential
equation with the graph of a particular solution. [The graphs
are labeled (a), (b), (c), and (d).] The correct match can be made
by comparing the frequency of the oscillations or the rate at
which the oscillations are being damped with the appropriate
coefficient in the differential equation.
(a) (b)
3
(c) (d)
3
y
y
1
2
2 3
y 9y 0
45. 46.
47. 48.
y 2y 10y 0
49. If the characteristic equation of the differential equation
y ay by 0
has two equal real roots given by m r, show that
y C 1 e rx C 2 xe rx
is a solution.
4
4
5 6
5 6
1
x
x
3
3
3
3
y
y
1
1
2 3
y y 37
y 25y 0
4
4 y 0
6
6
x
x
50. If the characteristic equation of the differential equation
y ay by 0
has complex roots given by m1 i and m2 i,
show that
y C 1e x cos x C 2e x sin x
is a solution.
True or False? In Exercises 51–54, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
51. y C1e is the general solution of
9 0.
3x C2e3x 52. is the general solution
of y4 y C1 C2xsin x C3 C4xcos x
2y y 0.
53. y x is a solution of
a0y 0 if and only if a1 a0 0.
54. It is possible to choose a and b such that y x is a solution
of
2ex y ay by 0.
The Wronskian of two differentiable functions f and g, denoted
by W( f, g), is defined as the function given by the determinant
W f, g f g g .
The functions f and g are linearly independent if there exists at
least one value of x for which W f, g 0. In Exercises 55–58,
use the Wronskian to verify the linear independence of the two
functions.
55. 56.
y 1 e ax
f
y 2 e bx , a b
57. y1 e 58. y1 x
ax sin bx
y 2 e ax cos bx, b 0
59. Euler’s differential equation is of the form
x 2 y axy by 0, x > 0
where a and b are constants.
(a) Show that this equation can be transformed into a secondorder
linear equation with constant coefficients by using the
substitution
(b) Solve x
60. Solve
2 x e
y 6xy 6y 0.
t .
y Ay 0
a n y n a n1 y n1 . . . a 1 y
y 1 e ax
y 2 xe ax
y 2 x 2
y 6y
where A is constant, subject to the conditions y0 0 and
y 0.