463 find the function u t dened for 0 2 and t 0

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Unformatted text preview: ine series of h2 (x). L Example. Suppose that we want to find the function u(x, t), defined for 0 ≤ x ≤ π and t ≥ 0, which satisfies the initial-value problem: ∂2u ∂2u = , ∂t2 ∂x2 u(0, t) = u(π, t) = 0, ∂u (x, 0) = 0. ∂t In this case, the first three coefficients for the Fourier sine series of h are u(x, 0) = 5 sin x + 12 sin 2x + 6 sin 3x, a1 = 5, a2 = 12, a3 = 6, and all the others are zero, so the solution must be u(x, t) = 5 sin x cos t + 12 sin 2x cos 2t + 6 sin 3x cos 3t. Exercises: 4.5.1 What happens to the frequency of the fundamental mode of oscillation of a vibrating string when the length of the string is doubled? When the tension on the string is doubled? When the density of the string is doubled? 4.5.2. Find the function u(x, t), defined for 0 ≤ x ≤ π and t ≥ 0, which satisfies the following conditions: ∂2u ∂2u = , ∂t2 ∂x2 u(0, t) = u(π, t) = 0, ∂u (x, 0) = 0. ∂t You may assume that the nontrivial solutions to the eigenvalue problem u(x, 0) = sin 2x, f (x) = λf (x), f (0) = 0 = f (π ) are...
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This document was uploaded on 01/12/2014.

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