Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 13.2, Problem 1P
After you find the series solution of a problem, make computer plots of your results as discussed just after equation (2.13).
Find the steady-state temperature distribution for the semi-infinite plate problem if the temperature of the bottom edge is
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Early Monday morning, the temperature in the lecture hall has fallen to 38°F, the same as the temperature
outside. At 7:00 A.M., the janitor turns on the furnace with the thermostat set at 70°F. The time constant for the
1 1
1
building is =3 hr and that for the building along with its heating system is K₁=3 hr. Assuming that the
outside temperature remains constant, what will be the temperature inside the lecture hall at 8:30 A.M.? When
will the temperature inside the hall reach 68°F?
At 8:30 A.M., the temperature inside the lecture hall will be about°F.
(Round to the nearest tenth as needed.)
In a hybrid-engine vehicle, energy from battery (“b" in Amperes) and from 95 octane unleaded gasoline
("g" in liters) are used intermittently to maximize the distance to be travelled (mileage) between fill-
ups and charge-ups. If mileage (M) in kilometers is found out to be:
M= 90 b+ 100 g - 3 b' - 5 g? - 2 b g
find the values of “b" and “g" that will maximize mileage M, find the maximum mileage and prove that
this is a maximum.
In modeling the effect of an impurity on crystal growth the following equation is used:
G-GL
Go-G KLCM
(1)
where C is the impurity concentration, Gi is a limiting growth rate, Go is the growth
rate of the crystal with no impurity present and KL and M are model parameters. The
table below shows the values of growth rates G for several impurity concentrations C,
for the values of Go = 310-3ml/min and G1 = 1.810-3ml/min.
75 100 125
C(ppm)
G(ml/min)x 103 2.5 2.2 2.04 1.95
50
Recall that in the multiple linear regression approach (generalized least squares method),
we write the general linear model as
y = coxo + C1x1+c2x2+ + CmXm
(2)
which is equivalent to
Ac = y,
where
X1
1
yi
1.
y2
(4)
1
Ут
Then we use the following linear algebra operation to find the parameters co, c1,
C2, *, Cm:
c= [A"A]*[A"y]
(5)
(a) Using the multiple linear regression approach (generalized least squares method),
write the model in the general linear form
y = coxo +C1x1+c2x2 +.+ CmXm
(6)
and then specify the form of…
Chapter 13 Solutions
Mathematical Methods in the Physical Sciences
Ch. 13.1 - Assume from electrostatics the equations E=/0 and...Ch. 13.1 - Show that the expression u=sin(xvt) describing a...Ch. 13.1 - Assume from electrodynamics the following...Ch. 13.1 - Obtain the heat flow equation (1.3) as follows:...Ch. 13.2 - After you find the series solution of a problem,...Ch. 13.2 - T=0,0x10,100,10x20. Solve the semi-infinite plate...Ch. 13.2 - Solve the semi-infinite plate problem if the...Ch. 13.2 - Solve the semi-infinite plate problem if the...Ch. 13.2 - Show that the solutions of (2.5) can also be...Ch. 13.2 - Show that the series in (2.12) can be summed to...
Ch. 13.2 - Solve Problem 3 if the plate is cut off at height...Ch. 13.2 - Find the steady-state temperature distribution in...Ch. 13.2 - Solve Problem 2 if the plate is cut off at height...Ch. 13.2 - Find the steady-state temperature distribution in...Ch. 13.2 - Find the steady-state temperature distribution in...Ch. 13.2 - Find the temperature distribution in a rectangular...Ch. 13.2 - Find the steady-state temperature distribution in...Ch. 13.2 - In the rectangular plate problem, we have so far...Ch. 13.2 - Consider a finite plate, 10cm by 30cm, with two...Ch. 13.2 - Show that there is only one function u which...Ch. 13.3 - Verify the coefficients in equation (3.14).Ch. 13.3 - A bar 10 cm long with insulated sides is initially...Ch. 13.3 - In the initial steady state of an infinite slab of...Ch. 13.3 - At t=0, two flat slabs each 5cm thick, one at 0...Ch. 13.3 - Prob. 5PCh. 13.3 - Show that the following problem is easily solved...Ch. 13.3 - A bar of length l with insulated sides has its...Ch. 13.3 - A bar of length 2 is initially at 0. From t=0 on,...Ch. 13.3 - Solve Problem 8 if, for t0, the x=0 end of the bar...Ch. 13.3 - Separate the wave equation (1.4) into a space...Ch. 13.3 - Solve the particle in a box problem to find (x,t)...Ch. 13.3 - Do Problem 11 if (x,0)=sin2x on (0,1).Ch. 13.4 - Complete the plucked string problem to get...Ch. 13.4 - A string of length l has a zero initial velocity...Ch. 13.4 - Solve Problem 2 if the initial displacement is:Ch. 13.4 - Solve Problem 2 if the initial displacement is :Ch. 13.4 - A string of length l is initially stretched...Ch. 13.4 - Do Problem 5 if the initial velocity V(x)=(y/t)t=0...Ch. 13.4 - Solve Problem 5 if the initial velocity is:Ch. 13.4 - Solve Problem 5 if the initial velocity is...Ch. 13.4 - In each of the Problems 1 to 8,find the frequency...Ch. 13.4 - Verify that, if k=nT, then the sum of the two...Ch. 13.4 - Verify (4.16) and find a similar formula for a...Ch. 13.4 - In Sections 2, 3, 4, we have solved a number of...Ch. 13.4 - Do Problem 12 for f(x)=1cos2x on (0,).Ch. 13.4 - Do Problem 12 for f(x)=xx3 on (0, 1).Ch. 13.5 - Compute numerically the coefficients (5.16) of the...Ch. 13.5 - Find the steady-state temperature distribution in...Ch. 13.5 - Find the steady-state temperature distribution in...Ch. 13.5 - A flat circular plate of radius a is initially at...Ch. 13.5 - Do Problem 4 if the initial temperature...Ch. 13.5 - Consider Problem 4 if the initial temperature...Ch. 13.5 - Find the steady-state temperature distribution in...Ch. 13.5 - Water at 100 is flowing through a long pipe of...Ch. 13.5 - Find the steady-state distribution of temperature...Ch. 13.5 - A cube is originally at 100. From t=0 on, the...Ch. 13.5 - The following two R(r) equations arise in various...Ch. 13.5 - Separate Laplaces equation in two dimensions in...Ch. 13.5 - Find the steady-state distribution of temperature...Ch. 13.5 - Find the steady state temperature distribution in...Ch. 13.5 - Solve Problem 14 if the temperatures of the two...Ch. 13.6 - Continue Figure 6.1 to show the fundamental modes...Ch. 13.6 - Prob. 2PCh. 13.6 - Separate the wave equation in two-dimensional...Ch. 13.6 - Find the characteristic frequencies for sound...Ch. 13.6 - A square membrane of side l is distorted into the...Ch. 13.6 - Let V=0 in the Schrödinger equation (3.22) and...Ch. 13.6 - In your Problem 6 solutions, find some examples of...Ch. 13.6 - Do Problem 6 in polar coordinates to find the...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Do Problem 11 if the curved surface is held at...Ch. 13.7 - Find the electrostatic potential outside a...Ch. 13.7 - Find the steady-state temperature distribution in...Ch. 13.7 - A sphere initially at 0 has its surface kept at...Ch. 13.7 - Separate the wave equation in spherical...Ch. 13.7 - Do Problem 6.6 in 3 dimensional rectangular...Ch. 13.7 - Prob. 18PCh. 13.7 - Find the eigenfunctions and energy eigenvalues for...Ch. 13.7 - Write the Schrödinger equation (3.22) if is a...Ch. 13.7 - Prob. 21PCh. 13.7 - Find the energy eigenvalues and eigen functions...Ch. 13.8 - Show that the gravitational potential V=Gm/r...Ch. 13.8 - Using the formulas of Chapter 12, Section 5, sum...Ch. 13.8 - Do the problem in Example 1 for the case of a...Ch. 13.8 - Prob. 4PCh. 13.8 - Find the method of images for problem 4.Ch. 13.8 - Substitute (8.25) into (8.22) and use (8.23) and...Ch. 13.8 - Verify that the Green function in (8.29) is zero...Ch. 13.8 - Show that the Green function (8.28) which is zero...Ch. 13.8 - Show that our results can be extended to find the...Ch. 13.9 - Verify that (9.15) follows from (9.14). Hint: Use...Ch. 13.9 - A metal plate covering the first quadrant has the...Ch. 13.9 - Consider the heat flow problem of Section 3. Solve...Ch. 13.9 - A semi-infinite bar is initially at temperature...Ch. 13.9 - Prob. 5PCh. 13.9 - Continue the problem of Example 2 in the following...Ch. 13.9 - Continue with Problem 4 as in Problem 6.Ch. 13.10 - Find the steady-state temperature distribution in...Ch. 13.10 - Solve Problem 1 if T=0 for x=0,x=1,y=0, and T=1x...Ch. 13.10 - Solve Problem 1 if the sides x=0 and x=1 are...Ch. 13.10 - Find the steady-state temperature distribution in...Ch. 13.10 - A bar of length l is initially at 0. From t=0 on,...Ch. 13.10 - Do Problem 5 if the x=0 end is insulated and the...Ch. 13.10 - Solve Problem 2 if the sides x=0 and x=1 are...Ch. 13.10 - A slab of thickness 10cm has its two faces at 10...Ch. 13.10 - A string of length l has initial displacement...Ch. 13.10 - Solve Problem 5.7 if half the curved surface of...Ch. 13.10 - The series in Problem 5.12 can be summed (see...Ch. 13.10 - A plate in the shape of a quarter circle has...Ch. 13.10 - Sum the series in Problem 12 to get...Ch. 13.10 - A long cylinder has been cut into quarter...Ch. 13.10 - Repeat Problems 12 and 13 for a plate in the shape...Ch. 13.10 - Consider the normal modes of vibration for a...Ch. 13.10 - Sketch some of the normal modes of vibration for a...Ch. 13.10 - Repeat Problem 17 for a membrane in the shape of a...Ch. 13.10 - Prob. 19MPCh. 13.10 - Use Problem 7.16 to find the characteristic...Ch. 13.10 - The surface temperature of a sphere of radius 1 is...Ch. 13.10 - Find the interior temperature in a hemisphere if...Ch. 13.10 - Find the steady-state temperature in the region...Ch. 13.10 - Find the general solution for the steady-state...Ch. 13.10 - The Klein-Gordon equation is 2u=1/v22u/t2+2u. This...Ch. 13.10 - Prob. 26MPCh. 13.10 - Do Problem 26 for a rectangular membrane.Ch. 13.10 - Find the steady-state temperature in a...
Additional Math Textbook Solutions
Find more solutions based on key concepts
The equivalent expression of x(y+z) by using the commutative property.
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
Assessment 1-1A Cookies are sold singly or in packages of 2 or 6. With this packaging, how many ways can you bu...
A Problem Solving Approach to Mathematics for Elementary School Teachers (12th Edition)
In a medical study, patients are classified in 8 ways according to whether they have blood type AB+, AB−, A+, A...
Probability and Statistics for Engineers and Scientists
Checkpoint1
Use the substitution method to solve this system:
Answers to Checkpoint exercises are found at the...
Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)
Solve each problem involving proportions. Price of Gasoline If 6 gallons of premium unleaded gasoline cost $17....
Mathematical Ideas (13th Edition) - Standalone book
56. Power Voting and Coalitions. Use the Web investigate the political coalitions at the national level in a pa...
Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)
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
- Find the unknown value. 27. y varies jointly with x and the cube root of 2. If when x=2 and z=27,y=12, find y if x=5 and z=8.arrow_forwardSuppose a certain country's population has constant relative birth and death rates of 97 births per thousand people per year and 47 deaths per thousand people per year respectively. Assume also that approximately 30,000 people emigrate (leave) from the country every year. Which equation best models the population P = P(t) of the country, where t is in years? dP A. dt = 50P + 30000 dP B. at C. D. **** dP at dP 0.05P - 30000 = 0.05P+ 30000 dt = 0.5P 30000 dp E. dt = 0.5P - 30arrow_forwardSolve.arrow_forward
- Please Help!arrow_forwardGraph solution of the O.D.Earrow_forwardThe ideal gas law pV = 0.082T shows the relationship among volume V, pressure p, and the temperature T for a fixed amount (1 mole) of a gas. But chemists believe that in many situations, the van der Waals equation gives more accurate results. If we measure temperature T in kelvins, volume V in liters, and pressure p in atmospheres (1 atm is the pressure exerted by the atmosphere at sea level), then the relationship for carbon dioxide is given by 0.082T 3.592 P = V – 0.043 atm. v2 What volume does this equation predict for 1 mole of carbon dioxide at 450 kelvins and 70 atm? (Suggestion: Consider volumes ranging from 0.1 to 1 liter. Round you answer.to.two.decimal places.)arrow_forward
- Early Monday morning, the temperature in the lecture hall has fallen to 41°F, the same as the temperature outside. At 7:00 A.M., the janitor turns on the furnace with the thermostat set at 72°F. The time constant for the building is 1 1 2 -= hr. K₁ 3 = 3 hr and that for the building along with its heating system is K Assuming that the outside temperature remains constant, what will be the temperature inside the lecture hall at 8:30 A.M.? When will the temperature inside the hall reach 67°F?arrow_forwardA measurement system can be modeled by the equation: 0.5y + y = F(t) Initially, the output signal is steady at 75 volts. The input signal is then suddenly increased to 100 volts. A. Determine the response equation. B. On the same graph, plot the input signal and the system time response through steady response.arrow_forwardH3.arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY
Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY