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D'Alembert's formula

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Utt means the second derivative with respect to t
Uxx means the second derivative with respect to x

Utt = 4Uxx, -(inf) < x < (inf), t > 0
U(x,0) = x,
Ut(x,0) = xe^(-x^2) for -(inf) < x < (inf)

Please use D'Alembert's Formula and show all work.
If there is Fourier series, please show how you got eigenvalues and eigenvectors. Also, please check your answer and make sure it is correct.
Thank you

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Solution Summary

This shows how to use D'Alembert's formula to solve an equation.

Solution Preview

Please see the attached doc file.
We have:

and we want to use D'Alembert's approach. Then let's consider:

note ...

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