Solved: Consider the wave equation a2uxx = utt in an infinite one-dimensional medium
Chapter 10, Problem 17(choose chapter or problem)
Consider the wave equation a2uxx = utt in an infinite one-dimensional medium subject to the initial conditions u( x, 0) = 0, ut ( x, 0) = g( x), < x < . a. Using the form of the solution obtained in 13, show that ( x) + ( x) = 0, a _( x) + a _( x) = g( x). b. Use the first equation of part a to show that _( x) = _( x). Then use the second equation to show that 2a_( x) = g( x) and therefore that ( x) = 1 2a _ x x0 g()d + ( x0), where x0 is arbitrary. Finally, determine ( x). c. Show that u( x, t) = 1 2a _ x+at xat g()d.
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