Differential Equations with Boundary-Value Problems (MindTap Course List)
9th Edition
ISBN: 9781305965799
Author: Dennis G. Zill
Publisher: Cengage Learning
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Textbook Question
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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A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the other end is at x = 3. The thermal diffusivity of the bar is k = 0.4. The bar’s initial temperature f(x) = 300 degrees Celsius. The ends of the bar (x = 0 and x = 3) are then put in an icy bath and kept at a constant 0 degrees C. Let u(x,t) be the temperature in the bar at x at time t, with t measured in seconds. Find u(x,t) and then u7(2,0.1).
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A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the other end is at x = 3. The thermal diffusivity of the
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Chapter 12 Solutions
Differential Equations with Boundary-Value Problems (MindTap Course List)
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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- 4. An engineer is planning a reinforced concrete ring on their computer (see picture below). The concrete ring has an inner radius r, outer radius R and height h. Such a ring has a volume of V = Th(R² – r²). The engineer is increasing the inner radius r and the height h by 1 m/sec at a certain time to. (a) What is the rate of change of the volume V at to if the engineer fixes R to have a constant value? (your answer should possibly involve r, R and h). (b) The engineer only has a fixed amount of concrete, so they fix V = 10500T m³. If r = 10 m, h = 20 m, what is the rate of change of R at to? Reinforced concrete ring - illustration.arrow_forward6. An unevenly heated metal plate has temperature T(x, y) in degrees Celsius at a point (x, y). If T(2, 1) = 119, Tz (2, 1) = 19, and Ty (2, 1) = -14, estimate the temperature at the point (2.04, 0.96).arrow_forward4. A 6-cm by 5-cm rectangular silver plate has being uniformly generated at each point at the rate 9 = 1.5 sec. Let x represent the distance along the edge of the plate of length 6 cm and y be the distance along the edge of the plate of length 5 cm. Suppose the temperature u along the edges is kept at cal cm³ the following temperatures: u(x,0) = x(6x), u(x, 5) = 0, u(0, y) = y(5 −y), u(6, y) = 0, 0 ≤ x ≤ 6, 0 ≤ y ≤ 5, where the origin lies at the corner of the plate with coordinates (0,0) and the edges lie along the positive x- and y-axes. The steady-state temperature u = u(x, y) satisfies Poisson's equation: Uxx (x, y) + Uyy (x, y) = q K' 0 < x < 6, 0 < y < 5, where K, the thermal conductivity, is 1.04 difference method with Ax = cal cm.deg.sec 3 and 4y = 2.5. Approximate the temperature u(x, y) using the finitearrow_forward
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