Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
expand_more
expand_more
format_list_bulleted
Textbook Question
Chapter 10.4, Problem 2E
In Problem 1-4, determine (a) the
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
The hyperbolic cosine function, designated by cosh x, is defined as cosh x = 1 /2 (ex + e-x ) (a) Show that f(x) = cosh x is an even function. (b) Graph f(x) = cosh x using a graphing utility. (c) Refer to Problem 128. Show that, for every x, (cosh x)2 - (sinh x)2 = 1
2. Suppose f(x+3)= f (x), what is the period of the function f?
(A) 4
(B) 3
(C) 2
(D) 1
Problem 1
Consider the function f defined on the interval (-3, 3) by
6+ 2x, -3 < x < 0,
f(x) =
10,
0 < x < 3,
and let g be the periodic-extension of f. That is, let g be the periodic function defined by g(x) = f(x),
-3 < a < 3, and g(x+6) = g(x).
(1) Find the Fourier Series for f. Include the main steps of the integrations in your solution.
(2) Sketch, on one set of axes, the graph of y = g(x) over the interval -9
Chapter 10 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 2ECh. 10.2 - Prob. 3ECh. 10.2 - Prob. 4ECh. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 6ECh. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...
Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 15-18, solve the heat flow problem...Ch. 10.2 - In Problems 15-18, solve the heat flow problem...Ch. 10.2 - In Problems 15-18, solve the heat flow problem...Ch. 10.2 - In Problems 15-18, solve the heat flow problem...Ch. 10.2 - In Problems 19-22, solve the vibrating string...Ch. 10.2 - In Problems 19-22, solve the vibrating string...Ch. 10.2 - In problem 19-22, solve the vibrating string...Ch. 10.2 - In problem 19-22, solve the vibrating string...Ch. 10.2 - Find the formal solution to the heat flow problem...Ch. 10.2 - Find the formal solution to the vibrating string...Ch. 10.2 - Prob. 25ECh. 10.2 - Verify that un(x,t) given in equation 10 satisfies...Ch. 10.2 - Prob. 27ECh. 10.2 - In Problems 27-30, a partial differential equation...Ch. 10.2 - Prob. 29ECh. 10.2 - In Problems 27-30, a partial differential equation...Ch. 10.2 - For the PDE in Problem 27, assume that the...Ch. 10.2 - For the PDE in Problem 29, assume the following...Ch. 10.2 - Prob. 33ECh. 10.3 - In Problems 1 -6, determine whether the given...Ch. 10.3 - In Problems 1 -6, determine whether the given...Ch. 10.3 - In Problems 1 -6, determine whether the given...Ch. 10.3 - In Problems 1 -6, determine whether the given...Ch. 10.3 - In Problems 1 -6, determine whether the given...Ch. 10.3 - In Problems 1 -6, determine whether the given...Ch. 10.3 - 7. Prove the following properties: a. If f and g...Ch. 10.3 - Verify the formula 5. Hint: Use the identity...Ch. 10.3 - In Problems 9-16, compute the Fourier series for...Ch. 10.3 - In Problems 9-16, compute the Fourier series for...Ch. 10.3 - In Problems 9-16, compute the Fourier series for...Ch. 10.3 - In Problems 9-16, compute the Fourier series for...Ch. 10.3 - In Problems 9-16, compute the Fourier series for...Ch. 10.3 - In Problems 17 -24, determine the function to...Ch. 10.3 - In Problems 17 -24, determine the function to...Ch. 10.3 - In Problems 17 -24, determine the function to...Ch. 10.3 - In Problems 17 -24, determine the function to...Ch. 10.3 - In Problems 17 -24, determine the function to...Ch. 10.3 - In Problems 17 -24, determine the function to...Ch. 10.3 - In Problems 17 -24, determine the function to...Ch. 10.3 - In Problems 17 -24, determine the function to...Ch. 10.3 - 25. Find the functions represented by the series...Ch. 10.3 - Show that the set of functions...Ch. 10.3 - Find the orthogonal expansion generalized Fourier...Ch. 10.3 - a. Show that the function f(x)=x2 has the Fourier...Ch. 10.3 - In Section 8.8, it was shown that the Legendre...Ch. 10.3 - As in Problem 29, find the first three...Ch. 10.3 - The Hermite polynomial Hn(x) are orthogonal on the...Ch. 10.3 - The Chebyshev Tchebichef polynomials Tn(x) are...Ch. 10.3 - Let {fn(x)} be an orthogonal set of functions on...Ch. 10.3 - Norm. The norm of a function f is like the length...Ch. 10.3 - Prob. 35ECh. 10.3 - Complex Form of the Fourier Series. a. Using the...Ch. 10.3 - Prob. 37ECh. 10.3 - Prob. 38ECh. 10.3 - Prob. 39ECh. 10.4 - In Problems 1-4, determine a the -periodic...Ch. 10.4 - In Problem 1-4, determine a the -periodic...Ch. 10.4 - In Problems 1-4, determine a the -periodic...Ch. 10.4 - In Problem 1-4, determine a the -periodic...Ch. 10.4 - In Problems 5 -10, compute the Fourier sine series...Ch. 10.4 - In Problems 5 -10, compute the Fourier sine series...Ch. 10.4 - In Problems 5 -10, compute the Fourier sine series...Ch. 10.4 - In Problems 5 -10, compute the Fourier sine series...Ch. 10.4 - In Problems 5 -10, compute the Fourier sine series...Ch. 10.4 - In Problems 5 -10, compute the Fourier sine series...Ch. 10.4 - In Problems 11 -16, compute the Fourier cosine...Ch. 10.4 - In Problems 11 -16, compute the Fourier cosine...Ch. 10.4 - In Problems 11 -16, compute the Fourier cosine...Ch. 10.4 - In Problems 11 -16, compute the Fourier cosine...Ch. 10.4 - In Problems 11 -16, compute the Fourier cosine...Ch. 10.4 - In Problems 11 -16, compute the Fourier cosine...Ch. 10.4 - In Problems 17 -19, for the given f(x), find the...Ch. 10.4 - In Problems 17 -19, for the given f(x), find the...Ch. 10.4 - In Problems 17 -19, for the given f(x), find the...Ch. 10.5 - In Problems 1 -10, find a formal solution to the...Ch. 10.5 - In Problems 1 -10, find a formal solution to the...Ch. 10.5 - Prob. 3ECh. 10.5 - Prob. 4ECh. 10.5 - Prob. 5ECh. 10.5 - In Problems 1 -10, find a formal solution to the...Ch. 10.5 - In Problems 1 -10, find a formal solution to the...Ch. 10.5 - Prob. 8ECh. 10.5 - Prob. 9ECh. 10.5 - In Problems 1-10, find a formal solution to the...Ch. 10.5 - Prob. 11ECh. 10.5 - Prob. 12ECh. 10.5 - Find a formal solution to the initial boundary...Ch. 10.5 - Prob. 14ECh. 10.5 - In Problems 15-18, find a formal solution to the...Ch. 10.5 - In Problems 15-18, find a formal solution to the...Ch. 10.5 - In Problems 15-18, find a formal solution to the...Ch. 10.5 - Prob. 18ECh. 10.5 - Prob. 19ECh. 10.6 - In Problems 1 -4, find a formal solution to the...Ch. 10.6 - Prob. 2ECh. 10.6 - Prob. 3ECh. 10.6 - Prob. 4ECh. 10.6 - The Plucked String. A vibrating string is governed...Ch. 10.6 - Prob. 6ECh. 10.6 - Prob. 7ECh. 10.6 - In Problems 7 and 8, find a formal solution to the...Ch. 10.6 - If one end of a string is held fixed while the...Ch. 10.6 - Derive a formula for the solution to the following...Ch. 10.6 - Prob. 11ECh. 10.6 - Prob. 12ECh. 10.6 - Prob. 13ECh. 10.6 - Prob. 14ECh. 10.6 - In Problems 13 -18, find the solution to the...Ch. 10.6 - In Problems 13 -18, find the solution to the...Ch. 10.6 - In Problems 13 -18, find the solution to the...Ch. 10.6 - In Problems 13 -18, find the solution to the...Ch. 10.6 - Derive the formal solution given in equation 22-24...Ch. 10.7 - In Problems 1-5, find a formal solution to the...Ch. 10.7 - Prob. 3ECh. 10.7 - In Problems 1-5, find a formal solution to the...Ch. 10.7 - Prob. 6ECh. 10.7 - In Problem 7 and8, find a solution to the...Ch. 10.7 - In Problems 7 and 8, find a solution to the...Ch. 10.7 - Find a solution to the Neumann boundary value...Ch. 10.7 - Prob. 13ECh. 10.7 - Prob. 15ECh. 10.7 - Prob. 16ECh. 10.7 - Prob. 18ECh. 10.7 - Prob. 19ECh. 10.7 - Stability.Use the maximum principle to prove the...Ch. 10.7 - Prob. 21E
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- Problem 2.1 (max, argmax, min, argmin) Let f(x) := (2+ sin(2.x)) and A := [0, 2]. (a) Compute: maxeA f(x) and argmax f(x). XEA (b) Compute: min,eg f(x) and argmin f(x). XEA Compute: argmax |[2 -e-f(x}+5 _ -가 (c) XEA (d) „Compute: argmin [11 – In (2 · f(x) + 5)].arrow_forwardProblem 3.7 Find the limits (a) lim,→0 sin 5x 3x (b) limə→0 Cos 0-1 20 sin 3x sin 5x (c) limr→0 r2 1arrow_forwardIn this problem you will use Rolle's theorem to determine whether it is possible for the function f(x) = - (7x' + 5x + 12) to have two or more real roots (or, equivalently, whether the graph of y = f(x) crosses the x-axis two or more times). Suppose that f(x) has at least two real roots. Choose two of these roots and call the smaller one a and the larger one b. By applying Rolle's theorem to f(x) on the interval [a, b], there exists at least one number e in the interval (a, b) so that f' (c) = The values of the derivative f' (x) = are always ? +, and therefore it is ? + for f(x) to have two or more real roots.arrow_forward
- In this problem you will use Rolle's theorem to determine whether it is possible for the function f(x) = - (7x' + 5x + 12) to have two or more real roots (or, equivalently, whether the graph of y = f(x) crosses the x- axis two or more times). Suppose that f(x) has at least two real roots. Choose two of these roots and call the smaller one a and the larger one b. By applying Rolle's theorem to f(x) on the interval [a, b], there exists at least one number c in the interval (a, b) so that f' (c) = The values of the derivative f' (x) = are always ? and therefore it is ? v for f(x) to have two or more real roots.arrow_forwardExample 2: Plot the function f(x)=cos3x below 0< <2arrow_forwardProblem 2. Sketch the graph of a function f(x) with f(0) = 0 that has the given sign charts for f'(x) and f"(x). 3, 2.5 1.5 1 0.5 -1.5 -1 -0.5 0.5 1.5 2.5 -0.5 f'(x): f"(x): 2.arrow_forward
- 2. (a) Let f be a function with domain R. Assume f has derivatives of every order. Find all possible real numbers A, B, C ER such that f(x) – [Ax² + Bx + C] lim x2 =0. (1) Note: In your answer, A, B and C will depend on values of f and its derivatives. We are asking for all possible answers. We want you to prove that your choices of A, B, and C satisfy (1), and that there are no other choices that satisfy (1).arrow_forward3. The water at a local beach has an average depth of 1 meter at low tide. The average depth of the water at high tide is 8 m. If one cycle takes 12 hours: (a) Determine the equation of this periodic function using cosine as the base function where o time is the beginning of high tide. (b) What is the depth of the water at 2 am? (c) Many people dive into the beach from the nearby dock. If the water must be at least 3 m deep to dive safely, between what daylight hours should people dive? Time (h) Height (m)arrow_forwardProblem #6: For which of the following can we use intermediate value theorem with the interval [a,b] = [0,2] to conclude that f(x) = 1 for some point in the interval (0, 2)? (i) f(x) = COS X 2 (ii) f(x) = 1-x (iii) f(x) = √√2-x (A) (i) only (B) none of them (C) (ii) and (iii) only (D) all of them (E) (iii) only (F) (i) and (iii) only (G) (i) and (ii) only (H) (ii) onlyarrow_forward
- Sketch a graph of the function f(x) 2 sin(2x). | -77/8-371/4-5/8-7/2 -37/8-rt/4-T/8 T/4 3t/8 /2 5n/8 37/4 7n/8 -2arrow_forwardSketch a graph of the function f(x) = sin(x). -7/4 1/2 37/4 57/4 37/2 77/4 元 -1 -2- . . .arrow_forwardSketch a graph of the function f(x) = cos(x). -7/4 1/2 37/4 57/4 Зд/2 77/4 -1 -2-arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
But what is the Fourier Transform? A visual introduction.; Author: 3Blue1Brown;https://www.youtube.com/watch?v=spUNpyF58BY;License: Standard YouTube License, CC-BY