Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
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Textbook Question
Chapter 4.6, Problem 2P
In each of Problems 1 through 4, write the given expression as a product of two trigonometric functions of different frequencies.
sin7t−sin4t
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Using FDF, BDF, and CDF, find the first derivative;
1. The distance x of a runner from a fixed point is measured (in meters) at an
interval of half a second. The data obtained is:
t
0
x
0
0.5
3.65
1.0
1.5
2.0
6.80
9.90
12.15
Use CDF to approximate the runner's velocity at times t = 0.5s and t = 1.5s
2. Using FDF, BDF, and CDF, find the first derivative of f(x)=x Inx for an input
of 2 assuming a step size of 1. Calculate using Analytical Solution and
Absolute Relative Error:
=
True Value - Approximate Value|
x100
True Value
3. Given the data below where f(x)
sin (3x), estimate f(1.5) using Langrage
Interpolation.
x
1
1.3
1.6
1.9
2.2
f(x)
0.14
-0.69
-0.99
-0.55
0.31
4. The vertical distance covered by a rocket from t=8 to t=30 seconds is given
by:
30
x =
Loo (2000ln
140000
140000
-
2100
9.8t) dt
Using the Trapezoidal Rule, n=2, find the distance covered.
5. Use Simpson's 1/3 and 3/8 Rule to approximate for sin x dx. Compare the
results for n=4 and n=8
Can you check if my step is correct?
I need help explaining on this example on how can I define the Time-Domain Function, Apply the Laplace Transformation Formula, and Simplify to Find the Frequency-Domain Expression. I need to understand on finding Y(s)
Chapter 4 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - A mass weighing stretches a spring . What is the...Ch. 4.1 - A mass attached to a vertical spring is slowly...Ch. 4.1 - A mass weighing stretches a spring . The mass is...
Ch. 4.1 - A mass of stretches a spring. The mass is set in...Ch. 4.1 - A mass weighing 3lb stretches a spring 3in. The...Ch. 4.1 - A series circuit has a capacitor of 0.25...Ch. 4.1 - A mass of stretches a spring . Suppose that the...Ch. 4.1 - A mass weighing 16lb stretches a spring 3in. The...Ch. 4.1 - A spring is stretched by a force of (N). A mass...Ch. 4.1 - A series circuit has a capacitor of 105farad, a...Ch. 4.1 - Suppose that a mass m slides without friction on a...Ch. 4.1 -
Duffing’s Equation
For the spring-mass system...Ch. 4.1 - A body of mass is attached between two springs...Ch. 4.1 - A cubic block of side and mass density per unit...Ch. 4.1 - In Problems through , we specift the mass, damping...Ch. 4.1 - In Problems 22 through 26, we specift the mass,...Ch. 4.1 - In Problems through , we specift the mass, damping...Ch. 4.1 - In Problems 22 through 26, we specift the mass,...Ch. 4.1 - In Problems 22 through 26, we specift the mass,...Ch. 4.1 - The Linear Versus the Nonlinear Pendulum.
Convert...Ch. 4.1 - (a) Numerical simulations as well as intuition...Ch. 4.2 - In each of the Problems 1 through 8, determine the...Ch. 4.2 - In each of the Problems through, determine the...Ch. 4.2 - In each of the Problems 1 through 8, determine the...Ch. 4.2 - In each of the Problems through, determine the...Ch. 4.2 - In each of the Problems 1 through 8, determine the...Ch. 4.2 - In each of the Problems through, determine the...Ch. 4.2 - In each of the Problems 1 through 8, determine the...Ch. 4.2 - In each of the Problems through, determine the...Ch. 4.2 - In each of the Problems through, find the...Ch. 4.2 - In each of the Problems through, find the...Ch. 4.2 - In each of the Problems through, find the...Ch. 4.2 - In each of the Problems 9 through 14, find the...Ch. 4.2 - In each of the Problems 9 through 14, find the...Ch. 4.2 - In each of the Problems through, find the...Ch. 4.2 - Verify that and are two solutions of the...Ch. 4.2 - Consider the differential operator T defined by...Ch. 4.2 - Can an equation y+p(t)y+q(t)y=0, with continuous...Ch. 4.2 - If the Wronskian W of f and g is 3e2t, and if...Ch. 4.2 - If the Wronskian W of f and g is t2et, and if...Ch. 4.2 - If W[f,g] is the Wronskian of f and g, and if...Ch. 4.2 - If the Wronskian of f and g is tcostsint, and if...Ch. 4.2 - In each of problem 22 through 25, verify that the...Ch. 4.2 - In each of problem 22 through 25, verify that the...Ch. 4.2 - In each of problem 22 through 25, verify that the...Ch. 4.2 - In each of problem 22 through 25, verify that the...Ch. 4.2 - 26. Consider the equation
(a). Show that and ...Ch. 4.2 - 27. Prove Theorem 4.2.4 and Corollary 4.2.5....Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - 37. The differential equation
Where N is...Ch. 4.2 - The differential equation y+(xy+y)=0 arises in the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26:
(a) Find the...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - Find a differential equation whose general...Ch. 4.3 - Find a differential equation whose general...Ch. 4.3 - Find a differential equation whose general...Ch. 4.3 - In each of Problems and , determine the values of...Ch. 4.3 - In each of Problems 47 and 48, determine the...Ch. 4.3 - If the roots of the characteristic equation are...Ch. 4.3 - Consider the equation ay+by+cy=d, where a,b,c and...Ch. 4.3 - Consider the equation , where and are constants...Ch. 4.3 - Prob. 52PCh. 4.3 - If , use the substitution to show that the...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems 54 through 61, find the...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems 54 through 61, find the...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems 54 through 61, find the...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems 62 through 65, find the...Ch. 4.3 - In each of Problems through, find the solution of...Ch. 4.3 - In each of Problems through, find the solution of...Ch. 4.3 - In each of Problems through, find the solution of...Ch. 4.4 - In each of Problems through , determine and so...Ch. 4.4 - In each of Problems through , determine and so...Ch. 4.4 - In each of Problems 1 through 4, determine 0,R,...Ch. 4.4 - In each of Problems 1 through 4, determine 0,R,...Ch. 4.4 - (a) A mass weighing lb stretches a spring in. If...Ch. 4.4 - (a) A mass of 100 g stretches a spring 5 cm. If...Ch. 4.4 - A mass weighing 3 lb stretches a spring 3 in. If...Ch. 4.4 - A series circuit has a capacitor of 0.25...Ch. 4.4 - (a) A mass of g stretches a spring cm. Suppose...Ch. 4.4 - A mass weighing 16 lb stretches a spring 3in. The...Ch. 4.4 - (a) A spring is stretched cm by a force of ...Ch. 4.4 - (a) A series circuit has a capacitor of farad, a...Ch. 4.4 - A certain vibrating system satisfies the equation...Ch. 4.4 - Show that the period of motion of an undamped...Ch. 4.4 - Show that the solution of the initial value...Ch. 4.4 - Show that Acos0t+Bsin0t can be written in the form...Ch. 4.4 - A mass weighing 8 lb stretches a spring 1.5 in....Ch. 4.4 - If a series circuit has a capacitor of C=0.8...Ch. 4.4 - Assume that the system described by the equation...Ch. 4.4 - Assume that the system described by the equation...Ch. 4.4 - Logarithmic Decrement For the damped oscillation...Ch. 4.4 - Referring to Problem , find the logarithmic...Ch. 4.4 - For the system in Problem , suppose that and ....Ch. 4.4 - The position of a certain spring-mass system...Ch. 4.4 - Consider the initial value problem . We wish to...Ch. 4.4 - Consider the initial value problem...Ch. 4.4 - Use the differential equation derived in Problem...Ch. 4.4 - Draw the phase portrait for the dynamical system...Ch. 4.4 - The position of a certain undamped spring-mass...Ch. 4.4 - The position of a certain spring-mass system...Ch. 4.4 - In the absence of damping, the motion of a...Ch. 4.4 - If the restoring force of a nonlinear spring...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30:
Determine a...Ch. 4.5 - In each of problems 23 through 30:
Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30:
Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - Consider the equation
(i)
From...Ch. 4.5 - Nonhomogeneous Cauchy-Euler Equations. In each of...Ch. 4.5 - Nonhomogeneous Cauchy-Euler Equations. In each of...Ch. 4.5 - Nonhomogeneous Cauchy-Euler Equations. In each of...Ch. 4.5 - Nonhomogeneous Cauchy-Euler Equations. In each of...Ch. 4.5 - Determine the general solution of
,
Where and ...Ch. 4.5 - In many physical problems, the nonhomogeneous term...Ch. 4.5 - Follow the instructions in Problem 37 to solve the...Ch. 4.6 - In each of Problems 1 through 4, write the given...Ch. 4.6 - In each of Problems 1 through 4, write the given...Ch. 4.6 - In each of Problems 1 through 4, write the given...Ch. 4.6 - In each of Problems 1 through 4, write the given...Ch. 4.6 - A mass weighing 4 pounds (lb) stretches a spring...Ch. 4.6 - A mass of 4 kg stretches a spring 8 cm. The mass...Ch. 4.6 - (a) Find the solution of Problem 5. (b) Plot the...Ch. 4.6 - 8.
Find the solution of the initial value problem...Ch. 4.6 - If an undamped spring-mass system with a mass that...Ch. 4.6 - A mass that weighs 8 lb stretches a spring 24 in....Ch. 4.6 - A spring is stretched 6 in. by a mass that weighs...Ch. 4.6 - A spring-mass system has a spring constant of 3...Ch. 4.6 - Furnish the details in determining when the gain...Ch. 4.6 - Find the solution of the initial value problem...Ch. 4.6 - A series circuit has a capacitor of 0.25...Ch. 4.6 - 16. Consider a vibrating system described by the...Ch. 4.6 - Consider the forced but undamped system described...Ch. 4.6 - Consider the vibrating system described by the...Ch. 4.6 - For the initial value problem in Problem 18, plot ...Ch. 4.6 - Problems 20 through 22 deal with the initial value...Ch. 4.6 - Problems 20 through 22 deal with the initial value...Ch. 4.6 - Problems 20 through 22 deal with the initial value...Ch. 4.6 - A spring-mass system with a hardening spring...Ch. 4.6 - Suppose that the system of Problem 23 is modified...Ch. 4.7 - (a) If
and ,
show that .
(b) Assuming that is...Ch. 4.7 - In each of Problems 2 through 5, use the method of...Ch. 4.7 - In each of Problems 2 through 5, use the method of...Ch. 4.7 - In each of Problems 2 through 5, use the method of...Ch. 4.7 - In each of Problems 2 through 5, use the method of...Ch. 4.7 - In each of Problems 6 through 9, find the solution...Ch. 4.7 - In each of Problems 6 through 9, find the solution...Ch. 4.7 - In each of Problems 6 through 9, find the solution...Ch. 4.7 - In each of Problems 6 through 9, find the solution...Ch. 4.7 - In each of Problems 10 through 13, use the method...Ch. 4.7 - In each of Problems 10 through 13, use the method...Ch. 4.7 - In each of Problems 10 through 13, use the method...Ch. 4.7 - In each of Problems 10 through 13, use the method...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 28 through 31, find the...Ch. 4.7 - In each of Problems 28 through 31, find the...Ch. 4.7 - In each of Problems 28 through 31, find the...Ch. 4.7 - In each of Problems 28 through 31, find the...Ch. 4.7 - Show that the solution of the initial value...Ch. 4.7 - By choosing the lower limit of integration in Eq....Ch. 4.7 - (a) Use the result of Problem 33 to show that...Ch. 4.7 - Use the result of Problem 33 to find the solution...Ch. 4.7 - Use the result of Problem 33 to find the...Ch. 4.7 - Use the result of Problem 33 to find the solution...Ch. 4.7 - By combining the results of the problems 35...Ch. 4.7 - The method of reduction of order (see the...Ch. 4.7 - In each of problems 40 and 41, use the method...Ch. 4.7 - In each of problems and , use the method outlined...Ch. 4.P1 - Denote by the displacement of the platform from...Ch. 4.P1 - Denote by the frequency response of , that is,...Ch. 4.P1 - Plot the graphs of versus the dimensionless ratio...Ch. 4.P1 - The vibrations in the floor of an industrial plant...Ch. 4.P1 - Test the results of your design strategy for the...Ch. 4.P2 - Show that the differential equation describing the...Ch. 4.P2 - (a) Find the linearization of at .
(b) In the...Ch. 4.P2 - Subject to the initial conditions , draw the graph...Ch. 4.P3 - Assuming that both springs have spring constant ...Ch. 4.P3 - The Heaviside, or unit step function, is defined...Ch. 4.P3 - Is the differential equation derived in Problems ...Ch. 4.P3 - In the case that the damping constant 0, find the...Ch. 4.P3 - Consider the case of an undamped problem using...Ch. 4.P3 - Consider the damped problem using the parameter...Ch. 4.P3 - Describe some other physical problems that could...Ch. 4.P4 - Problems 1 through 3 are concerned with one...Ch. 4.P4 - Problems 1 through 3 are concerned with one...Ch. 4.P4 - Problems 1 through 3 are concerned with one...Ch. 4.P4 - Problems and are concerned with systems that...Ch. 4.P4 - Problems and are concerned with systems that...Ch. 4.P4 - Carry out the calculations that lead from Eq. to...
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