Differential Equations with Boundary-Value Problems
9th Edition
ISBN: 9781337632515
Author: Dennis G. Zill
Publisher: Cengage Learning US
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Chapter 12.2, Problem 4E
In Problems 1–6 a rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t).
4. The ends are insulated, and there is heat transfer from the lateral surface into the surrounding medium at temperature 50. The initial temperature is 100 throughout.
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Chapter 12 Solutions
Differential Equations with Boundary-Value Problems
Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...
Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 18ECh. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 20ECh. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 22ECh. 12.1 - Prob. 23ECh. 12.1 - Prob. 24ECh. 12.1 - Prob. 25ECh. 12.1 - Prob. 26ECh. 12.1 - In Problems 27 and 28 show that the given partial...Ch. 12.1 - In Problems 27 and 28 show that the given partial...Ch. 12.1 - Verify that each of the products u = XY in (3),...Ch. 12.1 - Prob. 30ECh. 12.1 - Prob. 31ECh. 12.1 - Prob. 32ECh. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - Prob. 10ECh. 12.2 - In Problems 11 and 12 set up the boundary-value...Ch. 12.2 - In Problems 11 and 12 set up the boundary-value...Ch. 12.3 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.3 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.3 - Find the temperature u(x, t) in a rod of length L...Ch. 12.3 - Solve Problem 3 if L = 2 and f(x)={x,0x10,1x2.Ch. 12.3 - Suppose heat is lost from the lateral surface of a...Ch. 12.3 - Solve Problem 5 if the ends x = 0 and x = L are...Ch. 12.3 - A thin wire coinciding with the x-axis on the...Ch. 12.3 - Find the temperature u(x, t) for the...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - Prob. 11ECh. 12.4 - A model for the motion of a vibrating string whose...Ch. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - The transverse displacement u(x, t) of a vibrating...Ch. 12.4 - Prob. 19ECh. 12.4 - The vertical displacement u(x, t) of an infinitely...Ch. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 1–10 solve Laplace’s equation (1) for...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 1–10 solve Laplace’s equation (1) for...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - Prob. 10ECh. 12.5 - In Problems 11 and 12 solve Laplaces equation (1)...Ch. 12.5 - In Problems 11 and 12 solve Laplaces equation (1)...Ch. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - In Problems 15 and 16 use the superposition...Ch. 12.5 - In Problems 15 and 16 use the superposition...Ch. 12.5 - Prob. 18ECh. 12.5 - Solve the Neumann problem for a rectangle:...Ch. 12.5 - Prob. 20ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 3ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 6ECh. 12.6 - Prob. 7ECh. 12.6 - Prob. 8ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - In Problems 13-16 proceed as in Example 2 to solve...Ch. 12.6 - Prob. 15ECh. 12.6 - In Problems 13-16 proceed as in Example 2 to solve...Ch. 12.6 - Prob. 17ECh. 12.6 - Prob. 18ECh. 12.6 - Prob. 19ECh. 12.6 - Prob. 20ECh. 12.7 - In Example 1 find the temperature u(x, t) when the...Ch. 12.7 - Prob. 2ECh. 12.7 - Find the steady-state temperature for a...Ch. 12.7 - Prob. 4ECh. 12.7 - Prob. 5ECh. 12.7 - Prob. 6ECh. 12.7 - Prob. 7ECh. 12.7 - Prob. 8ECh. 12.7 - Prob. 9ECh. 12.7 - Prob. 10ECh. 12.8 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.8 - Prob. 2ECh. 12.8 - Prob. 3ECh. 12.8 - In Problems 3 and 4 solve the wave equation (2)...Ch. 12.8 - Prob. 5ECh. 12.8 - Prob. 6ECh. 12 - Use separation of variables to find product...Ch. 12 - Use separation of variables to find product...Ch. 12 - Find a steady-state solution (x) of the...Ch. 12 - Give a physical interpretation for the boundary...Ch. 12 - At t = 0 a string of unit length is stretched on...Ch. 12 - Prob. 6RECh. 12 - Find the steady-state temperature u(x, y) in the...Ch. 12 - Find the steady-state temperature u(x, y) in the...Ch. 12 - Prob. 9RECh. 12 - Find the temperature u(x, t) in the infinite plate...Ch. 12 - Prob. 11RECh. 12 - Solve the boundary-value problem 2ux2+sinx=ut, 0 ...Ch. 12 - Prob. 13RECh. 12 - The concentration c(x, t) of a substance that both...Ch. 12 - Prob. 15RECh. 12 - Solve Laplaces equation for a rectangular plate...Ch. 12 - Prob. 17RECh. 12 - Prob. 18RECh. 12 - Prob. 19RECh. 12 - If the four edges of the rectangular plate in...
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