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(a) Obtain product solutions, = f(x)g(y), of (7.2.14) that satisfy = 0 on the four sides
Chapter 7, Problem 7.3.1(choose chapter or problem)
(a) Obtain product solutions, = f(x)g(y), of (7.2.14) that satisfy = 0 on the four sides of a rectangle. (Hint: If necessary, see Section 7.3.) (b) Using part (a), solve the initial value problem for a vibrating rectangular membrane (fixed on all sides). (c) Using part (a), solve the initial value problem for the two-dimensional heat equation with zero temperature on all sides.Solve the initial value problem and analyze the temperature as t if the boundary conditions are * (a) u(0, y, t) = 0, u(L, y, t) = 0, u(x, 0, t) = 0, u(x, H, t)=0 (b) u x(0, y, t) = 0, u x(L, y, t) = 0, u y (x, 0, t) = 0, u y (x, H, t)=0 * (c) u x(0, y, t) = 0, u x(L, y, t) = 0, u(x, 0, t) = 0, u(x, H, t)=0 (d) u(0, y, t) = 0, u x(L, y, t) = 0, u y (x, 0, t) = 0, u y (x, H, t)=0 (e) u(0, y, t) = 0, u(L, y, t) = 0, u(x, 0, t) = 0, u y (x, H, t) + hu(x, H, t)=0 (h > 0)
Questions & Answers
QUESTION:
(a) Obtain product solutions, = f(x)g(y), of (7.2.14) that satisfy = 0 on the four sides of a rectangle. (Hint: If necessary, see Section 7.3.) (b) Using part (a), solve the initial value problem for a vibrating rectangular membrane (fixed on all sides). (c) Using part (a), solve the initial value problem for the two-dimensional heat equation with zero temperature on all sides.Solve the initial value problem and analyze the temperature as t if the boundary conditions are * (a) u(0, y, t) = 0, u(L, y, t) = 0, u(x, 0, t) = 0, u(x, H, t)=0 (b) u x(0, y, t) = 0, u x(L, y, t) = 0, u y (x, 0, t) = 0, u y (x, H, t)=0 * (c) u x(0, y, t) = 0, u x(L, y, t) = 0, u(x, 0, t) = 0, u(x, H, t)=0 (d) u(0, y, t) = 0, u x(L, y, t) = 0, u y (x, 0, t) = 0, u y (x, H, t)=0 (e) u(0, y, t) = 0, u(L, y, t) = 0, u(x, 0, t) = 0, u y (x, H, t) + hu(x, H, t)=0 (h > 0)
ANSWER:Step 1 of 6
let us seek solutions of this form
Consider the partial differential equation
where
