Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 7.10, Problem 1P
In Problems 1 to 3, the graphs sketched represent one period of the excess pressure
1.
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3. Verify that the indicated function (or family of functions) is a solution of the given differential equation. Assume an
appropriate interval I of definition for each solution.
The average number of minutes Americans commute to work is 27.7 minutes (Sterling's Best Places, April 13, 2012). The average commute time in minutes for 48 cities are as follows:
Click on the datafile logo to reference the data.
DATA file
Albuquerque
23.3
Jacksonville
26.2
Phoenix
28.3
Atlanta
28.3
Kansas City
23.4
Pittsburgh
25.0
Austin
24.6
Las Vegas
28.4
Portland
26.4
Baltimore
32.1
Little Rock
20.1
Providence
23.6
Boston
31.7
Los Angeles
32.2
Richmond
23.4
Charlotte
25.8
Louisville
21.4
Sacramento
25.8
Chicago
38.1
Memphis
23.8
Salt Lake City
20.2
Cincinnati
24.9
Miami
30.7
San Antonio
26.1
Cleveland
26.8
Milwaukee
24.8
San Diego
24.8
Columbus
23.4
Minneapolis
23.6
San Francisco
32.6
Dallas
28.5
Nashville
25.3
San Jose
28.5
Denver
28.1
New Orleans
31.7
Seattle
27.3
Detroit
29.3
New York
43.8
St. Louis
26.8
El Paso
24.4
Oklahoma City
22.0
Tucson
24.0
Fresno
23.0
Orlando
27.1
Tulsa
20.1
Indianapolis
24.8
Philadelphia
34.2
Washington, D.C.
32.8
a. What is the mean commute time for…
(b)
313
dy
dx
-y= 10 sin(2x)y;
y(x) = ex-5 cos(2x)
Chapter 7 Solutions
Mathematical Methods in the Physical Sciences
Ch. 7.2 - In Problems 1 to 6 find the amplitude, period,...Ch. 7.2 - In Problems 1 to 6 find the amplitude, period,...Ch. 7.2 - In Problems 1 to 6 find the amplitude, period,...Ch. 7.2 - In Problems 1 to 6 find the amplitude, period,...Ch. 7.2 - In Problems 1 to 6 find the amplitude, period,...Ch. 7.2 - In Problems 1 to 6 find the amplitude, period,...Ch. 7.2 - In Problems 7 to 10 you are given a complex...Ch. 7.2 - In Problems 7 to 10 you are given a complex...Ch. 7.2 - In Problems 7 to 10 you are given a complex...Ch. 7.2 - In Problems 7 to 10 you are given a complex...
Ch. 7.2 - The charge q on a capacitor in a simple a-c...Ch. 7.2 - RepeatProblem11:(a)ifq=Re4e30it;(b)ifq=Im4e30it.Ch. 7.2 - A simple pendulum consists of a point mass m...Ch. 7.2 - The displacements x of two simple pendulums (see...Ch. 7.2 - As in Problem 14, the displacements x of two...Ch. 7.2 - As in Problem 14, let the displacements be...Ch. 7.2 - Show that equation (2.10) for a wave can be...Ch. 7.2 - In Problems 18 to 20, find the amplitude, period,...Ch. 7.2 - In Problems 18 to 20, find the amplitude, period,...Ch. 7.2 - In Problems 18 to 20, find the amplitude, period,...Ch. 7.2 - Write the equation for a sinusoidal wave of...Ch. 7.2 - Do Problem 21 for a wave of amplitude 4, period 6,...Ch. 7.2 - Write an equation for a sinusoidal sound wave of...Ch. 7.2 - The velocity of sound in sea water is about...Ch. 7.2 - Write an equation for a sinusoidal radio wave of...Ch. 7.3 - For each of the following combinations of a...Ch. 7.3 - For each of the following combinations of a...Ch. 7.3 - For each of the following combinations of a...Ch. 7.3 - For each of the following combinations of a...Ch. 7.3 - Using the definition (end of Section 2) of a...Ch. 7.3 - In Problems 6 and 7, use a trigonometry formula to...Ch. 7.3 - In Problems 6 and 7, use a trigonometry formula to...Ch. 7.3 - A periodic modulated (AM) radio signal has the...Ch. 7.4 - Show that if f(x) has period p, the average value...Ch. 7.4 - (a) Prove that 0/2sin2xdx=0/2cos2xdx by making the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - In Problems 3 to 12, find the average value of the...Ch. 7.4 - Using (4.3) and equations similar to (4.5) to...Ch. 7.4 - Use the results of Problem 13 to evaluate the...Ch. 7.4 - Use the results of Problem 13 to evaluate the...Ch. 7.4 - Use the results of Problem 13 to evaluate the...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - In each of the following problems you are given a...Ch. 7.5 - Show that in (5.2) the average values of...Ch. 7.5 - Write out the details of the derivation of...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - For each of the periodic functions in Problems 5.1...Ch. 7.6 - Use a computer to produce graphs like Fig. 6.2...Ch. 7.6 - Repeat the example using the same Fourier series...Ch. 7.6 - Use Problem 5.7 to show that oddn1/n2=2/8. Try...Ch. 7.6 - UseProblem5.11toshowthat1221+1421+1621+=12.Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Expand the same functions as in Problems 5.1 to...Ch. 7.7 - Show that if a real f(x) is expanded in a complex...Ch. 7.7 - If f(x)=12a0+1ancosnx+1bnsinnx=cneinx, use Eulers...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - In Problems 5.1 to 5.9, define each function by...Ch. 7.8 - (a) Sketch several periods of the function f(x) of...Ch. 7.8 - In Problems 11 to 14, parts (a) and (b), you are...Ch. 7.8 - In Problems 11 to 14, parts (a) and (b), you are...Ch. 7.8 - In Problems 11 to 14, parts (a) and (b), you are...Ch. 7.8 - In Problems 11 to 14, parts (a) and (b), you are...Ch. 7.8 - Sketch (or computer plot) each of the following...Ch. 7.8 - Each of the following functions is given over one...Ch. 7.8 - Each of the following functions is given over one...Ch. 7.8 - Each of the following functions is given over one...Ch. 7.8 - Each of the following functions is given over one...Ch. 7.8 - Each of the following functions is given over one...Ch. 7.8 - Write out the details of the derivation of the...Ch. 7.9 - The functions in Problems 1 to 3 are neither even...Ch. 7.9 - The functions in Problems 1 to 3 are neither even...Ch. 7.9 - The functions in Problems 1 to 3 are neither even...Ch. 7.9 - The functions in Problems 1 to 3 are neither even...Ch. 7.9 - Each of the functions in Problems 5 to 12 is given...Ch. 7.9 - Each of the functions in Problems 5 to 12 is given...Ch. 7.9 - Each of the functions in Problems 5 to 12 is given...Ch. 7.9 - Each of the functions in Problems 5 to 12 is given...Ch. 7.9 - Each of the functions in Problems 5 to 12 is given...Ch. 7.9 - Each of the functions in Problems 5 to 12 is given...Ch. 7.9 - Each of the functions in Problems 5 to 12 is given...Ch. 7.9 - Each of the functions in Problems 5 to 12 is given...Ch. 7.9 - Give algebraic proofs of (9.3). Hint: Write...Ch. 7.9 - Give algebraic proofs that for even and odd...Ch. 7.9 - Given f(x)=x for 0x1, sketch the even function fc...Ch. 7.9 - Let f(x)=sin2x,0x. Sketch (or computer plot) the...Ch. 7.9 - In Problems 17 to 22 you are given f(x) on an...Ch. 7.9 - In Problems 17 to 22 you are given f(x) on an...Ch. 7.9 - In Problems 17 to 22 you are given f(x) on an...Ch. 7.9 - In Problems 17 to 22 you are given f(x) on an...Ch. 7.9 - In Problems 17 to 22 you are given f(x) on an...Ch. 7.9 - In Problems 17 to 22 you are given f(x) on an...Ch. 7.9 - If a violin string is plucked (pulled aside and...Ch. 7.9 - If, in Problem 23, the string is stopped at the...Ch. 7.9 - Suppose that f(x) and its derivative f(x) are both...Ch. 7.9 - In Problems 26 and 27, find the indicated Fourier...Ch. 7.9 - In Problems 26 and 27, find the indicated Fourier...Ch. 7.10 - In Problems 1 to 3, the graphs sketched represent...Ch. 7.10 - In Problems 1 to 3, the graphs sketched represent...Ch. 7.10 - In Problems 1 to 3, the graphs sketched represent...Ch. 7.10 - In Problems 4 to 10, the sketches show several...Ch. 7.10 - In Problems 4 to 10, the sketches show several...Ch. 7.10 - In Problems 4 to 10, the sketches show several...Ch. 7.10 - In Problems 4 to 10, the sketches show several...Ch. 7.10 - In Problems 4 to 10, the sketches show several...Ch. 7.10 - In Problems 4 to 10, the sketches show several...Ch. 7.10 - In Problems 4 to 10, the sketches show several...Ch. 7.11 - Prove (11.4) for a function of period 2l expanded...Ch. 7.11 - Prove that if f(x)=i=cneinx, then the average...Ch. 7.11 - If f(x) is complex, we usually want the average of...Ch. 7.11 - When a current I flows through a resistance R, the...Ch. 7.11 - Use Parsevals theorem and the results of the...Ch. 7.11 - Use Parsevals theorem and the results of the...Ch. 7.11 - Use Parsevals theorem and the results of the...Ch. 7.11 - Use Parsevals theorem and the results of the...Ch. 7.11 - Use Parsevals theorem and the results of the...Ch. 7.11 - A general form of Parsevals theorem says that if...Ch. 7.11 - Let f(x) on (0,2l) satisfy f(2lx)=f(x), that is,...Ch. 7.12 - Following a method similar to that used in...Ch. 7.12 - Do Example 1 above by using a cosine transform...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 3 to 12, find the exponential Fourier...Ch. 7.12 - In Problems 13 to 16, find the Fourier cosine...Ch. 7.12 - In Problems 13 to 16, find the Fourier cosine...Ch. 7.12 - In Problems 13 to 16, find the Fourier cosine...Ch. 7.12 - In Problems 13 to 16, find the Fourier cosine...Ch. 7.12 - In Problems 17 to 20, find the Fourier sine...Ch. 7.12 - In Problems 17 to 20, find the Fourier sine...Ch. 7.12 - In Problems 17 to 20, find the Fourier sine...Ch. 7.12 - In Problems 17 to 20, find the Fourier sine...Ch. 7.12 - Find the Fourier transform of f(x)=ex2/22. Hint:...Ch. 7.12 - The function j1()=(cossin)/ is of interest in...Ch. 7.12 - Using Problem 17, show that...Ch. 7.12 - (a) Find the exponential Fourier transform of...Ch. 7.12 - (a) Represent as an exponential Fourier transform...Ch. 7.12 - Using Problem 15, show that 01cos2d=2.Ch. 7.12 - Represent each of the following functions (a) by a...Ch. 7.12 - Represent each of the following functions (a) by a...Ch. 7.12 - Represent each of the following functions (a) by a...Ch. 7.12 - Represent each of the following functions (a) by a...Ch. 7.12 - Verify Parsevals theorem (12.24) for the special...Ch. 7.12 - Verify Parsevals theorem (12.24) for the special...Ch. 7.12 - Verify Parsevals theorem (12.24) for the special...Ch. 7.12 - Show that if (12.2) is written with the factor 1/2...Ch. 7.12 - Starting with the symmetrized integrals as in...Ch. 7.12 - Normalize f(x) in Problem 21; that is find the...Ch. 7.13 - The displacement (from equilibrium) of a particle...Ch. 7.13 - The symbol [x] means the greatest integer less...Ch. 7.13 - We have said that Fourier series can represent...Ch. 7.13 - The diagram shows a relaxation oscillator. The...Ch. 7.13 - Consider one arch of f(x)=sinx. Show that the...Ch. 7.13 - Let f(t)=eit on (,). Expand f(t) in a complex...Ch. 7.13 - Given f(x)=x on (,), expand f(x) in an appropriate...Ch. 7.13 - From facts you know, find in your head the average...Ch. 7.13 - Given f(x)= x,0x1, 2,1x2. (a) Sketch at least...Ch. 7.13 - (a) Sketch at least three periods of the graph of...Ch. 7.13 - Find the three Fourier series in Problems 9 and...Ch. 7.13 - What would be the apparent frequency of a sound...Ch. 7.13 - (a) Given f(x)=(x)/2 on (0,), find the sine series...Ch. 7.13 - (a) Find the Fourier series of period 2 for...Ch. 7.13 - Given f(x)=1,2x0,1,0x2, find the exponential...Ch. 7.13 - Given f(x)=x,0x1,2x,1x2,0,x2, find the cosine...Ch. 7.13 - Show that the Fourier sine transform of x1/2 is...Ch. 7.13 - Let f(x) and g() be a pair of Fourier transforms....Ch. 7.13 - Find the form of Parsevals theorem ( 12.24) for...Ch. 7.13 - Find the exponential Fourier transform of...Ch. 7.13 - Define a function h(x)=k=f(x+2k), assuming that...Ch. 7.13 - Use Poissons formula (Problem 21b) and Problem 20...Ch. 7.13 - Use Parsevals theorem and Problem 12.11 to...
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