Differential Equations and Linear Algebra (4th Edition)
4th Edition
ISBN: 9780321964670
Author: Stephen W. Goode, Scott A. Annin
Publisher: PEARSON
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Chapter 8.5, Problem 1P
To determine
The circular frequency of the system, the amplitude, phase and period of the motion.
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A mass weighing 16 pounds is attached to a spring, stretching it 2 feet. A damping
mechanism provides a resistance numerically equal to b times the instantaneous velocity.
The mass is pulled down 1 foot below equilibrium and released from rest.
a. Find the equations of motion for b = 3, b = 4, b = 5.
b. Determine if each system is underdamped, critically damped, or overdamped.
Suppose an object weighing 64 pounds stretches a spring 8 feet. If the object is attached to the spring and released 5 feet below the equilibrium position from rest, find the equation of motion of the object x(t).
4. A car supported by a MacPherson strut (shock absorber system) travels on a bumpy
road at a constant velocity v. The equation modeling the motion of the car is
Tut
80x + 1000x
= 2500 cos
where r = x (t) represents the vertical position of the cars axle relative to its equilib-
rium position, and the basic units of measurement are feet and feet per second (this
is actually just an example of a forced, un-damped harmonic oscillator, if that is any
help). The constant numbers above are related to the characteristics of the car and
the strut. Note that the coefficient of time t (inside the cosine) in the forcing term
on the right hand side is a frequency, which in this case is directly proportional to the
velocity v of the car.
(a) Find the general solution to this nonhomogeneous ODE. Note that your answer
will have a term in it which is a function of v.
Chapter 8 Solutions
Differential Equations and Linear Algebra (4th Edition)
Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - Prob. 5TFRCh. 8.1 - Prob. 6TFRCh. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - Prob. 8TFRCh. 8.1 - Prob. 9TFRCh. 8.1 - True-False Review For Questions a-j, decide if the...
Ch. 8.1 - Problems For problem 1-4, find Ly for the given...Ch. 8.1 - Problems For problem 1-4, find Ly for the given...Ch. 8.1 - Problems For problem 1-4, find Ly for the given...Ch. 8.1 - Problems For problem 1-4, find Ly for the given...Ch. 8.1 - Problems For problem 5-9, verify that given...Ch. 8.1 - Prob. 6PCh. 8.1 - Problems For problem 5-9, verify that given...Ch. 8.1 - Problems For problem 5-9, verify that given...Ch. 8.1 - Prob. 9PCh. 8.1 - Problems For problem 10-13, compute Ker(L). L=D3x2Ch. 8.1 - For Problems 10-13, compute Ker(L). L=D2+1Ch. 8.1 - For Problem 10-13, compute ker(L). L=D2+2D15Hint:...Ch. 8.1 - For Problems 10-13, compute Ker(L). L=x2D+x.Ch. 8.1 - Problems For problems 14-15, find L1L2 and L2L1...Ch. 8.1 - Problems For problems 14-15, find L1L2 and L2L1...Ch. 8.1 - Problems If L1=D+a1(x), determine all differential...Ch. 8.1 - Problems For Problems 17-18, write the given...Ch. 8.1 - Prob. 18PCh. 8.1 - Prob. 19PCh. 8.1 - Prob. 20PCh. 8.1 - Determine which of the following sets of vectors...Ch. 8.1 - Determine which of the following sets of vectors...Ch. 8.1 - For Problems 23-26, determine two linearly...Ch. 8.1 - For Problems 23-26, determine two linearly...Ch. 8.1 - For Problems 23-26, determine two linearly...Ch. 8.1 - For Problems 23-26, determine two linearly...Ch. 8.1 - For Problems 27-31, determine three linearly...Ch. 8.1 - For Problems 27-31, determine three linearly...Ch. 8.1 - For Problems 27-31, determine three linearly...Ch. 8.1 - Prob. 30PCh. 8.1 - For Problems 27-31, determine three linearly...Ch. 8.1 - Prob. 32PCh. 8.1 - For Problems 32-33, determine four linearly...Ch. 8.1 - For Problems 34-35, determine two linearly...Ch. 8.1 - For Problems 34-35, determine two linearly...Ch. 8.1 - For Problems 34-35, determine three linearly...Ch. 8.1 - Prob. 37PCh. 8.1 - Determine a particular solution for the...Ch. 8.1 - Determine a particular solution for the...Ch. 8.1 - Prob. 40PCh. 8.1 - Determine a particular solution for the...Ch. 8.1 - Prob. 42PCh. 8.1 - Prob. 43PCh. 8.1 - Extend the proof of theorem 8.1.4 to an arbitrary...Ch. 8.1 - Prob. 45PCh. 8.1 - Let T:VW be a linear transformation, and suppose...Ch. 8.2 - True-False Review For Questions a-h, decide if the...Ch. 8.2 - True-False Review For Questions a-h, decide if the...Ch. 8.2 - Prob. 3TFRCh. 8.2 - Prob. 4TFRCh. 8.2 - True-False Review For Questions a-h, decide if the...Ch. 8.2 - Prob. 6TFRCh. 8.2 - Prob. 7TFRCh. 8.2 - Prob. 8TFRCh. 8.2 - Problems For Problems 1-3, determine a basis for...Ch. 8.2 - Prob. 2PCh. 8.2 - Problems For Problems 1-3, determine a basis for...Ch. 8.2 - Prob. 4PCh. 8.2 - Prob. 5PCh. 8.2 - Determine a basis for the solution space to y+4y=0...Ch. 8.2 - Determine a basis for the solution space to y+y=0...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 8-34, determine the general...Ch. 8.2 - Problems For Problems 35-38, solve the given...Ch. 8.2 - Problems For Problems 35-38, solve the given...Ch. 8.2 - Problems For Problems 35-38, solve the given...Ch. 8.2 - Problems For Problems 35-38, solve the given...Ch. 8.2 - Prob. 39PCh. 8.2 - Prob. 40PCh. 8.2 - Prob. 41PCh. 8.2 - Prob. 42PCh. 8.2 - Prob. 43PCh. 8.2 - Problems Consider P(D)=y. What conditions on the...Ch. 8.2 - Prob. 45PCh. 8.2 - Prob. 46PCh. 8.2 - Prob. 47PCh. 8.2 - Prob. 48PCh. 8.2 - Prob. 49PCh. 8.2 - Prob. 50PCh. 8.2 - Prob. 51PCh. 8.2 - Prob. 54PCh. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problem 1-16, determine the annihilator of the...Ch. 8.3 - For Problems, 17-32, determine the general...Ch. 8.3 - For Problems, 17-32, determine the general...Ch. 8.3 - For Problems, 17-32, determine the general...Ch. 8.3 - For Problems, 17-32, determine the general...Ch. 8.3 - For Problems, 17-32, determine the general...Ch. 8.3 - For Problems, 17-32, determine the general...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problems 17-32, determine the general solution...Ch. 8.3 - For Problem 33-37, solve the given initial value...Ch. 8.3 - For Problem 33-37, solve the given initial value...Ch. 8.3 - For Problems 33-37, solve the given initial value...Ch. 8.3 - For Problems 33-37, solve the given initial value...Ch. 8.3 - For Problems 33-37, solve the given initial value...Ch. 8.3 - At time t the displacement from equilibrium, y(t),...Ch. 8.3 - For Problems 39-47, determine an appropriate trial...Ch. 8.3 - For Problems 39-47, determine an appropriate trial...Ch. 8.3 - For Problems 39-47, determine an appropriate trial...Ch. 8.3 - For Problems 39-47, determine an appropriate trial...Ch. 8.3 - For Problems 39-47, determine an appropriate trial...Ch. 8.3 - Problems For problems 39-47, determine an...Ch. 8.3 - Prob. 45PCh. 8.3 - Problems For problems 39-47, determine an...Ch. 8.3 - Problems For problems 39-47, determine an...Ch. 8.3 - Problems Derive an appropriate trial solution for...Ch. 8.5 - For Questions ai, decide if the given statement is...Ch. 8.5 - For Questions ai, decide if the given statement is...Ch. 8.5 - For Questions ai, decide if the given statement is...Ch. 8.5 - Prob. 4TFRCh. 8.5 - For Questions ai, decide if the given statement is...Ch. 8.5 - Prob. 6TFRCh. 8.5 - Prob. 7TFRCh. 8.5 - Prob. 8TFRCh. 8.5 - Prob. 9TFRCh. 8.5 - Prob. 1PCh. 8.5 - Prob. 2PCh. 8.5 - A force of 3N stretches a spring by 1m. a Find the...Ch. 8.5 - For Problems 4-10, determine the motion of the...Ch. 8.5 - For Problems 4-10, determine the motion of the...Ch. 8.5 - For Problems 4-10, determine the motion of the...Ch. 8.5 - For Problems 4-10, determine the motion of the...Ch. 8.5 - For Problems 4-10, determine the motion of the...Ch. 8.5 - For Problems 4-10, determine the motion of the...Ch. 8.5 - Prob. 10PCh. 8.5 - In the previous problem, find the time at which...Ch. 8.5 - Consider the spring-mass system whose motion is...Ch. 8.5 - Consider the spring-mass system whose motion is...Ch. 8.5 - Consider the spring-mass system whose motion is...Ch. 8.5 - Prob. 15PCh. 8.5 - Prob. 16PCh. 8.5 - Prob. 18PCh. 8.5 - Prob. 19PCh. 8.5 - Prob. 20PCh. 8.5 - Prob. 21PCh. 8.5 - An object of mass m is attached to the midpoint of...Ch. 8.5 - Repeat the previous problem if the string has...Ch. 8.5 - Consider the damped spring-mass system whose...Ch. 8.5 - Consider the spring-mass system whose motion is...Ch. 8.5 - Consider the spring-mass system whose motion is...Ch. 8.5 - Prob. 27PCh. 8.5 - The general solution of forced undamped...Ch. 8.5 - Determine the period of the motion for the...Ch. 8.5 - Prob. 30PCh. 8.5 - Prob. 31PCh. 8.5 - Consider the damped spring-mass system with m=1,...Ch. 8.5 - Consider the spring-mass system whose motion is...Ch. 8.5 - Prob. 34PCh. 8.6 - For Questions af, decide if the given statement is...Ch. 8.6 - Prob. 2TFRCh. 8.6 - Prob. 3TFRCh. 8.6 - For Questions af, decide if the given statement is...Ch. 8.6 - Prob. 5TFRCh. 8.6 - Prob. 6TFRCh. 8.6 - Problems Determine the charge on the capacitor at...Ch. 8.6 - Problems Determine the steady current in the RLC...Ch. 8.6 - Prob. 3PCh. 8.6 - Prob. 4PCh. 8.6 - Consider the RLC circuit with R=16, L=8H, C=140F,...Ch. 8.6 - Show that the differential equation governing the...Ch. 8.6 - Prob. 7PCh. 8.6 - Prob. 8PCh. 8.7 - Problems For Problems 122, use the variation of...Ch. 8.7 - Problems For Problems 122, use the variation of...Ch. 8.7 - Problems For Problems 122, use the variation of...Ch. 8.7 - Problems For Problems 122, use the variation of...Ch. 8.7 - Problems For Problems 122, use the variation of...Ch. 8.7 - Problems For Problems 122, use the variation of...Ch. 8.7 - Problems For Problems 122, use the variation of...Ch. 8.7 - Problems For Problems 122, use the variation of...Ch. 8.7 - Problems For Problems 122, use the...Ch. 8.7 - Problems For Problems 122, use the...Ch. 8.7 - Problems For Problems 122, use the...Ch. 8.7 - Prob. 12PCh. 8.7 - Prob. 13PCh. 8.7 - Problems For Problems 122, use the...Ch. 8.7 - Problems
For Problems 1-22, use the...Ch. 8.7 - Problems
For Problems 1-22, use the...Ch. 8.7 - Prob. 17PCh. 8.7 - Prob. 18PCh. 8.7 - Prob. 19PCh. 8.7 - Prob. 20PCh. 8.7 - Prob. 21PCh. 8.7 - Prob. 22PCh. 8.7 - Problems
For Problems 23-26, use a Greens...Ch. 8.7 - Problems
For Problems 23-26, use a Greens...Ch. 8.7 - Problems For problems 23-26, use a Greens function...Ch. 8.7 - Problems For problems 23-26, use a Greens function...Ch. 8.7 - Problems For problem 27-28, use a Greens function...Ch. 8.7 - Prob. 28PCh. 8.7 - Prob. 29PCh. 8.7 - Problems Consider the differential equation...Ch. 8.7 - Prob. 31PCh. 8.7 - Prob. 32PCh. 8.7 - Prob. 33PCh. 8.7 - Prob. 34PCh. 8.7 - Prob. 35PCh. 8.7 - Prob. 36PCh. 8.7 - Prob. 37PCh. 8.8 - Problems For Problems 1-8, determine the general...Ch. 8.8 - Problems For Problems 1-8, determine the general...Ch. 8.8 - Problems For Problems 1-8, determine the general...Ch. 8.8 - Problems For Problems 1-8, determine the general...Ch. 8.8 - Problems For Problems 1-8, determine the general...Ch. 8.8 - Problems For Problems 1-8, determine the general...Ch. 8.8 - Problems For Problems 1-8, determine the general...Ch. 8.8 - Problems For Problems 1-8, determine the general...Ch. 8.8 - Problems For Problems 9-11, find the solution to...Ch. 8.8 - Prob. 10PCh. 8.8 - Prob. 11PCh. 8.8 - Problems
Consider the Cauchy-Euler equation
...Ch. 8.8 - Problems
For Problems 14-21, solve the given...Ch. 8.8 - Problems
For Problems 14-21, solve the given...Ch. 8.8 - Problems
For Problems 14-21, solve the given...Ch. 8.8 - Problems
For Problems 14-21, solve the given...Ch. 8.8 - Prob. 18PCh. 8.8 - Problems
For Problems 14-21, solve the given...Ch. 8.8 - Prob. 20PCh. 8.8 - Prob. 21PCh. 8.8 - Prob. 24PCh. 8.9 - For Problems 16, y1 is a solution to the given...Ch. 8.9 - For Problems 16, y1 is a solution to the given...Ch. 8.9 - For Problems 16, y1 is a solution to the given...Ch. 8.9 - For Problems 16, y1 is a solution to the given...Ch. 8.9 - For Problems 16, y1 is a solution to the given...Ch. 8.9 - For Problems 16, y1 is a solution to the given...Ch. 8.9 - Consider the Cauchy-Euler equation...Ch. 8.9 - Prob. 8PCh. 8.9 - Prob. 9PCh. 8.9 - For Problems 1015, y1 is a solution to the...Ch. 8.9 - For Problems 1015, y1 is a solution to the...Ch. 8.9 - Prob. 12PCh. 8.9 - Prob. 13PCh. 8.9 - Prob. 14PCh. 8.9 - For Problems 1015, y1 is a solution to the...Ch. 8.9 - Consider the differential equation...Ch. 8.10 - In Problems 1-6, find Ly for the given...Ch. 8.10 - Prob. 2APCh. 8.10 - Prob. 3APCh. 8.10 - Prob. 4APCh. 8.10 - Prob. 5APCh. 8.10 - Prob. 6APCh. 8.10 - Prob. 7APCh. 8.10 - Prob. 8APCh. 8.10 - Prob. 9APCh. 8.10 - Prob. 10APCh. 8.10 - Prob. 11APCh. 8.10 - Prob. 12APCh. 8.10 - Prob. 13APCh. 8.10 - In Problem 14-17, determine the annihilator of the...Ch. 8.10 - Prob. 15APCh. 8.10 - Prob. 16APCh. 8.10 - Prob. 17APCh. 8.10 - Prob. 18APCh. 8.10 - In Problem 18-23, determine a trial solution for...Ch. 8.10 - Prob. 20APCh. 8.10 - Prob. 21APCh. 8.10 - Prob. 22APCh. 8.10 - Prob. 23APCh. 8.10 - For Problems 24-29, solve the given nonhomogeneous...Ch. 8.10 - Prob. 25APCh. 8.10 - For Problems 24-29, solve the given nonhomogeneous...Ch. 8.10 - Prob. 27APCh. 8.10 - Prob. 28APCh. 8.10 - Prob. 29APCh. 8.10 - Prob. 30APCh. 8.10 - Prob. 31APCh. 8.10 - Prob. 32APCh. 8.10 - For Problem 30-39, state whether the annihilator...Ch. 8.10 - Prob. 34APCh. 8.10 - For Problem 30-39, state whether the annihilator...Ch. 8.10 - For Problem 30-39, state whether the annihilator...Ch. 8.10 - Prob. 37APCh. 8.10 - Prob. 38APCh. 8.10 - For Problem 30-39, state whether the annihilator...Ch. 8.10 - Prob. 40APCh. 8.10 - For Problems 40-45, use the annihilator to solve...Ch. 8.10 - Prob. 42APCh. 8.10 - Prob. 43APCh. 8.10 - Prob. 44APCh. 8.10 - Prob. 45APCh. 8.10 - Prob. 46APCh. 8.10 - Prob. 47APCh. 8.10 - For Problem 47-51, use the variations of...Ch. 8.10 - Prob. 49APCh. 8.10 - Prob. 50APCh. 8.10 - Prob. 51APCh. 8.10 - Prob. 52APCh. 8.10 - Prob. 53APCh. 8.10 - In the Problem 53-58, find the general solution to...Ch. 8.10 - Prob. 55APCh. 8.10 - In the Problem 53-58, find the general solution to...Ch. 8.10 - Prob. 57APCh. 8.10 - Prob. 58APCh. 8.10 - For Problems 59-61, solve the differential...Ch. 8.10 - Prob. 60APCh. 8.10 - Prob. 61APCh. 8.10 - Prob. 62APCh. 8.10 - Prob. 63APCh. 8.10 - Prob. 64APCh. 8.10 - Prob. 65APCh. 8.10 - Prob. 66AP
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