3. Consider the initial valve problem for U = U(t, x) for XER given by Ut + UU x = 0, U (0,x) = 1+ x² Write down a formula for the characteristic line that goes through the point y on the x-axis. sketch some of the characteristic lines in the (t,x)- plane calculate the critical time, t*, at which the solution, u, develops a vertical tangent line. Use the method of characteristics to sketch the Solution as a function of X just before and just after the critical time (Just after the critical time interpret the solution as a multivalved function)

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A:

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A:

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A:

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A:

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A:

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A: