Question: 5 (3pts). Consider the wave equation utt=uxx for all 0 with the initial conditions u(x,0)=ex and ut(x,0)=0. Using the Fourier transform, show that the wave

5 (3pts). Consider the wave equation utt=uxx for all 0 with

5 (3pts). Consider the wave equation utt=uxx for all 0 with the initial conditions u(x,0)=ex and ut(x,0)=0. Using the Fourier transform, show that the wave equation has a solution of the form u(x,t)=21f(w)eiwxcos(wt)dw, where f(w) denotes the Fourier transform of ex. Prove that this solution also satisfies u(x,t)=201+w2cos(wt)cos(wx)dw

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