Consider the Boundary-Initial Value problem 8²u 9 მ2 u(0, t) = 0, u(x, 0) = x (3 — x), ²u 0 < x <3, Ət2, u(3, t) = 0, t> 0 du -(x,0) = 0, 00 This models the displacement u(x, t) of a freely vibrating string, with fixed ends, initial profile x (3x), and zero initial velocity. The solution u(x, t), is given by the series u(x, t) = Σbn sin (nx) cos(₂ t) 3 n=1

Consider the Boundary-Initial Value problem 8²u 9 მ2 u(0, t) = 0, u(x, 0) = x (3 — x), ²u 0 < x <3, Ət2, u(3, t) = 0, t> 0 du -(x,0) = 0, 00 This models the displacement u(x, t) of a freely vibrating string, with fixed ends, initial profile x (3x), and zero initial velocity. The solution u(x, t), is given by the series u(x, t) = Σbn sin (nx) cos(₂ t) 3 n=1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Transcribed Image Text:Consider the Boundary-Initial Value problem 8²u 9 მ2 u(0, t) = 0, u(x,0) = x(3 - x), where 9 b₁ = (1-(-1)") bn 8²u Ət²¹ and Cn = 0 < x < 3, u(x, t) = This models the displacement u(x, t) of a freely vibrating string, with fixed ends, initial profile x (3 – x), and zero initial velocity. The solution u(x, t), is given by the series t> 0 u(3, t) = 0, du t> 0 - (x,0) = = 0, 0<x<3 Ət ㅠ 42 b, sin (n. 7 2) cos(C₂ t) bn n=1

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