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(b) Let U(x,t) denote a solution to the wave equation Uµ – c²uxx = 0. Show that V(x, t) = U(ax, at) is also a solution to the wave equation for any constant a. (c) Find the solution to the problem Ut - 2Uxx = 0, x € R, 1 U(x,0) = 1+ x2' Ut(x,0) = 0. Provide a sketch of the solution for different times.

(b) Let U(x,t) denote a solution to the wave equation Uµ – c²uxx = 0. Show that V(x, t) = U(ax, at) is also a solution to the wave equation for any constant a. (c) Find the solution to the problem Ut - 2Uxx = 0, x € R, 1 U(x,0) = 1+ x2' Ut(x,0) = 0. Provide a sketch of the solution for different times.

Advanced Engineering Mathematics
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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(b) Let U(x,t) denote a solution to the wave equation U – c²uxx = 0. Show that V (x, t) = U(ax, at) is also a solution to the wave equation for any constant a. (c) Find the solution to the problem U – 2Uxx = 0, x € R, 1 U(x,0) = 1+ x2' U;(x,0) = 0. Provide a sketch of the solution for different times.
Transcribed Image Text:(b) Let U(x,t) denote a solution to the wave equation U – c²uxx = 0. Show that V (x, t) = U(ax, at) is also a solution to the wave equation for any constant a. (c) Find the solution to the problem U – 2Uxx = 0, x € R, 1 U(x,0) = 1+ x2' U;(x,0) = 0. Provide a sketch of the solution for different times.
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