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 5.1, Problem 1P
In each of Problems 1 through 4, sketch the graph of the given function. In each case, determine whether f is continuous, piecewise continuous, or neither on the interval 0≤t≤3.
f(t) = {t2, 0 ≤ t ≤ 12 + t, 1 < t ≤ 2 6 − t, 2 < t ≤ 3
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Students have asked these similar questions
5. Suppose that a mass of 5 stretches a spring 10. The mass is acted on by an external force
of F(t)=10 sin () and moves in a medium that gives a damping coefficient of ½. If the mass
is set in motion with an initial velocity of 3 and is stretched initially to a length of 5. (I
purposefully removed the units- don't worry about them. Assume no conversions are
needed.)
a) Find the equation for the displacement of the spring mass at time t.
b) Write the equation for the displacement of the spring mass in phase-mode form.
c) Characterize the damping of the spring mass system as overdamped, underdamped or
critically damped. Explain how you know.
D.E. for Spring Mass Systems
k
m* g = kLo
y" +—y' + — —±y = —±F(t), y(0) = yo, y'(0) = vo
m
2
A₁ = √c₁² + C₂²
Q = tan-1
4. Given the following information determine the appropriate trial solution to find yp. Do not
solve the differential equation. Do not find the constants.
a) (D-4)2(D+ 2)y = 4e-2x
b) (D+ 1)(D² + 10D +34)y = 2e-5x cos 3x
9.7 Given the equations
0.5x₁-x2=-9.5
1.02x₁ - 2x2 = -18.8
(a) Solve graphically.
(b) Compute the determinant.
(c) On the basis of (a) and (b), what would you expect regarding
the system's condition?
(d) Solve by the elimination of unknowns.
(e) Solve again, but with a modified slightly to 0.52. Interpret
your results.
Chapter 5 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Ch. 5.1 - In each of Problems 1 through 4, sketch the graph...Ch. 5.1 - In each of Problems through , sketch the graph of...Ch. 5.1 - In each of Problems 1 through 4, sketch the graph...Ch. 5.1 - In each of Problems 1 through 4, sketch the graph...Ch. 5.1 - In each of Problems through , determine whether...Ch. 5.1 - In each of Problems through , determine whether...Ch. 5.1 - In each of Problems 5 through 12, determine...Ch. 5.1 - In each of Problems through , determine whether...Ch. 5.1 - In each of Problems through , determine whether...Ch. 5.1 - In each of Problems through , determine whether...
Ch. 5.1 - In each of Problems 5 through 12, determine...Ch. 5.1 - In each of Problems 5 through 12, determine...Ch. 5.1 -
Find the Laplace transform of each of the...Ch. 5.1 - In each of Problems 14 through 17, find the...Ch. 5.1 - In each of Problems through , find the Laplace...Ch. 5.1 - In each of Problems through , find the Laplace...Ch. 5.1 - In each of Problems 14 through 17, find the...Ch. 5.1 - In each of Problems through , find the Laplace...Ch. 5.1 - In each of Problems through , find the Laplace...Ch. 5.1 - In each of Problems 18 through 21, find the...Ch. 5.1 - In each of Problems through , find the Laplace...Ch. 5.1 - In each of Problems 22 through 24, use the facts...Ch. 5.1 - In each of Problems 22 through 24, use the facts...Ch. 5.1 - In each of Problems through , use the facts that ...Ch. 5.1 - In each of Problems through , using integration...Ch. 5.1 - In each of Problems 25 through 30, using...Ch. 5.1 - In each of Problems through , using integration...Ch. 5.1 - In each of Problems through , using integration...Ch. 5.1 - In each of Problems through , using integration...Ch. 5.1 - In each of Problems 25 through 30, using...Ch. 5.1 - In each of Problems 31 through 34, determine...Ch. 5.1 - In each of Problems 31 through 34, determine...Ch. 5.1 - In each of Problems 31 through 34, determine...Ch. 5.1 - In each of Problems through , determine whether...Ch. 5.1 - A Proof of Corollary 5.1.7 (a) Starting from...Ch. 5.1 - The Gamma Function. The gamma function is defined...Ch. 5.1 - Consider the laplace transform of tp, where p1....Ch. 5.2 - In each of Problems through, find the Laplace...Ch. 5.2 - In each of Problems through, find the Laplace...Ch. 5.2 - In each of Problems 1 through 10, find the Laplace...Ch. 5.2 - In each of Problems through, find the Laplace...Ch. 5.2 - In each of Problems through, find the Laplace...Ch. 5.2 - In each of Problems through, find the Laplace...Ch. 5.2 - In each of Problems 1 through 10, find the Laplace...Ch. 5.2 - In each of Problems through, find the Laplace...Ch. 5.2 - In each of Problems 1 through 10, find the Laplace...Ch. 5.2 - In each of Problems through, find the Laplace...Ch. 5.2 - (a) Let F(s)=L{f(t)}, where f(t) is piecewise...Ch. 5.2 - In each of Problems through, transform the given...Ch. 5.2 - In each of Problems through, transform the given...Ch. 5.2 - In each of Problems 12 through 21, transform the...Ch. 5.2 - In each of Problems 12 through 21, transform the...Ch. 5.2 - In each of Problems 12 through 21, transform the...Ch. 5.2 - In each of Problems 12 through 21, transform the...Ch. 5.2 - In each of Problems through, transform the given...Ch. 5.2 - In each of Problems through, transform the given...Ch. 5.2 - In each of Problems 12 through 21, transform the...Ch. 5.2 - In each of Problems 12 through 21, transform the...Ch. 5.2 - In section 4.1 the differential equation for the...Ch. 5.2 - In each of Problems through, find the Laplace...Ch. 5.2 - In each of Problems 23 through 27, find the...Ch. 5.2 - In each of Problems 23 through 27, find the...Ch. 5.2 - In each of Problems 23 through 27, find the...Ch. 5.2 - In each of Problems 23 through 27, find the...Ch. 5.2 - The Laplace transforms of certain functions can be...Ch. 5.2 - For each of the following initial value problems,...Ch. 5.3 - In each of Problems 1 through 8, find the unknown...Ch. 5.3 - In each of Problems 1 through 8, find the unknown...Ch. 5.3 - In each of Problems 1 through 8, find the unknown...Ch. 5.3 - In each of Problems 1 through 8, find the unknown...Ch. 5.3 - In each of Problems 1 through 8, find the unknown...Ch. 5.3 - In each of Problems 1 through 8, find the unknown...Ch. 5.3 - In each of Problems 1 through 8, find the unknown...Ch. 5.3 - In each of Problems 1 through 8, find the unknown...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 9 through 24, using the...Ch. 5.3 - In each of Problems 25 through 28, use a computer...Ch. 5.3 - In each of Problems 25 through 28, use a computer...Ch. 5.3 - In each of Problems 25 through 28, use a computer...Ch. 5.3 - In each of Problems 25 through 28, use a computer...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems 1 through 13, use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems 1 through 13, use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems 14 through 19, use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems 14 through 19, use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems 14 through 19, use the Laplace...Ch. 5.4 - In each of Problems through , use the Laplace...Ch. 5.4 - In each of Problems through , use a computer...Ch. 5.4 - In each of Problems 20 through 24, use a computer...Ch. 5.4 - In each of Problems through , use a computer...Ch. 5.4 - In each of Problems 20 through 24, use a computer...Ch. 5.4 - In each of Problems 20 through 24, use a computer...Ch. 5.4 - Use the Laplace transform to solve the system...Ch. 5.4 - A radioactive substance R1 having decay rate k1...Ch. 5.5 - In each of Problems through , sketch the graph of...Ch. 5.5 - In each of Problems 1 through 6, sketch the graph...Ch. 5.5 - In each of Problems 1 through 6, sketch the graph...Ch. 5.5 - In each of Problems through , sketch the graph of...Ch. 5.5 - In each of Problems through , sketch the graph of...Ch. 5.5 - In each of Problems through , sketch the graph of...Ch. 5.5 - In each of Problems 7 through 12, find the Laplace...Ch. 5.5 - In each of Problems 7 through 12, find the Laplace...Ch. 5.5 - In each of Problems through , find the Laplace...Ch. 5.5 - In each of Problems through , find the Laplace...Ch. 5.5 - In each of Problems through , find the Laplace...Ch. 5.5 - In each of Problems through , find the Laplace...Ch. 5.5 - In each of Problems 13 through 18, find the...Ch. 5.5 - In each of Problems through , find the inverse...Ch. 5.5 - In each of Problems 13 through 18, find the...Ch. 5.5 - In each of Problems 13 through 18, find the...Ch. 5.5 - In each of Problems through , find the inverse...Ch. 5.5 - In each of Problems 13 through 18, find the...Ch. 5.5 - In each of Problems through , find the Laplace...Ch. 5.5 - In each of Problems 19 through 21, find the...Ch. 5.5 - In each of Problems through , find the Laplace...Ch. 5.5 - In each of Problems through , find the Laplace...Ch. 5.5 - In each of Problems through , find the Laplace...Ch. 5.5 - In each of Problems 22 through 24, find the...Ch. 5.5 - (a) If f(t)=1u1(t), find L{f(t)}; compare with...Ch. 5.5 - Consider the function defined by
and has...Ch. 5.6 - In each of Problems 1 through 13, find the...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - In each of Problems 1 through 13, find the...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - In each of Problems 1 through 13, find the...Ch. 5.6 - In each of Problems 1 through 13, find the...Ch. 5.6 - In each of Problems through , find the solutions...Ch. 5.6 - Find an expression involving uc(t) for a function...Ch. 5.6 - Find an expression involving for a function that...Ch. 5.6 - A certain spring-mass system satisfies the initial...Ch. 5.6 - Modify the problem in Example 1 of this section by...Ch. 5.6 - Consider the initial value problem y+13y+4y=fk(t),...Ch. 5.6 - In Problem 19 through 23, we explore the effect of...Ch. 5.6 - In Problem 19 through 23, we explore the effect of...Ch. 5.6 - In Problem 19 through 23, we explore the effect of...Ch. 5.6 - In Problem 19 through 23, we explore the effect of...Ch. 5.6 - In Problem 19 through 23, we explore the effect of...Ch. 5.7 - In each of Problems through , find the solutions...Ch. 5.7 - In each of Problems through , find the solutions...Ch. 5.7 - In each of Problems 1 through 12, find the...Ch. 5.7 - In each of Problems through , find the solutions...Ch. 5.7 - In each of Problems 1 through 12, find the...Ch. 5.7 - In each of Problems through , find the solutions...Ch. 5.7 - In each of Problems 1 through 12, find the...Ch. 5.7 - In each of Problems 1 through 12, find the...Ch. 5.7 - In each of Problems through , find the solutions...Ch. 5.7 - In each of Problems 1 through 12, find the...Ch. 5.7 - In each of Problems through , find the solutions...Ch. 5.7 - In each of Problems through , find the solutions...Ch. 5.7 - Consider again the system in Example 3 of this...Ch. 5.7 - Consider the initial value problem y+y+y=(t1),...Ch. 5.7 - Consider the initial value problem
Where ...Ch. 5.7 - Consider the initial value problem
Where ...Ch. 5.7 - Problem 17 through 22 deal with effect of a...Ch. 5.7 - Problem 17 through 22 deal with effect of a...Ch. 5.7 - Problem 17 through 22 deal with effect of a...Ch. 5.7 - Problem 17 through 22 deal with effect of a...Ch. 5.7 - Problem 17 through 22 deal with effect of a...Ch. 5.7 - Problem 17 through 22 deal with effect of a...Ch. 5.7 - The position of a certain lightly damped...Ch. 5.7 - Proceed as in Problem 23 for the oscillator...Ch. 5.7 - a) By the method of variation of parameters, show...Ch. 5.8 - Establish the distributive and associative...Ch. 5.8 - Show, by means of the example f(t)=sint, that ff...Ch. 5.8 - In each of Problems 3 through 6, find the Laplace...Ch. 5.8 - In each of Problems 3 through 6, find the Laplace...Ch. 5.8 - In each of Problems through , find the Laplace...Ch. 5.8 - In each of Problems through , find the Laplace...Ch. 5.8 - In each of Problems 7 through 12, find the inverse...Ch. 5.8 - In each of Problems 7 through 12, find the inverse...Ch. 5.8 - In each of Problems through , find the inverse...Ch. 5.8 - In each of Problems through , find the inverse...Ch. 5.8 - In each of Problems 7 through 12, find the inverse...Ch. 5.8 - In each of Problems 7 through 12, find the inverse...Ch. 5.8 - (a) If f(t)=tm and g(t)=tn, where m and n are...Ch. 5.8 - In each of Problems through , express the total...Ch. 5.8 - In each of Problems through , express the total...Ch. 5.8 - In each of Problems through , express the total...Ch. 5.8 - In each of Problems through , express the total...Ch. 5.8 - In each of Problems 14 through 21, express the...Ch. 5.8 - In each of Problems through , express the total...Ch. 5.8 - In each of Problems 14 through 21, express the...Ch. 5.8 - In each of Problems 14 through 21, express the...Ch. 5.8 - Unit Step Responses. The unit step response of a...Ch. 5.8 - Consider the equation (t)+0tk(t)()d=f(t), in which...Ch. 5.8 - Consider the Volterra integral equation (see...Ch. 5.8 - In each of Problems 25 through 27:
Solve the given...Ch. 5.8 - In each of Problems 25 through 27: a) Solve the...Ch. 5.8 - In each of Problems 25 through 27:
Solve the given...Ch. 5.8 - There are also equations, known as...Ch. 5.8 - There are also equations, known as...Ch. 5.8 - There are also equations, known as...Ch. 5.8 - The Tautochrone. A problem of interest in the...Ch. 5.9 - Find the transfer function of the system shown in...Ch. 5.9 - Find the transfer function of the system shown in...Ch. 5.9 - If h(t) is any one of the functions,...Ch. 5.9 - Prob. 4PCh. 5.9 - Use Rouths criterion to find necessary and...Ch. 5.9 - For each of the characteristic function in...Ch. 5.9 - For each of the characteristic function in...Ch. 5.9 - For each of the characteristic function in...Ch. 5.9 - For each of the characteristic function in...Ch. 5.9 - For each of the characteristic function in...Ch. 5.9 - For each of the characteristic function in...Ch. 5.9 - In each of Problems 12 through 15, use the Routh...Ch. 5.9 - In each of Problems 12 through 15, use the Routh...Ch. 5.9 - In each of Problems through, use the Routh...Ch. 5.9 - In each of Problems through, use the Routh...Ch. 5.P1 - Find such that the Laplace transform of the...Ch. 5.P1 - Suppose that the impressed voltage is prescribed...Ch. 5.P1 - Suppose that the impressed voltage is prescribed ...Ch. 5.P2 - In the following problem, we ask the reader to...Ch. 5.P2 - In the following problem, we ask the reader to...Ch. 5.P2 - In the following problem, we ask the reader to...Ch. 5.P2 - In the following problem, we ask the reader to...Ch. 5.P2 - In the following problem, we ask the reader to...Ch. 5.P2 - In the following problem, we ask the reader to...Ch. 5.P2 - In the following problem, we ask the reader to...
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