11 04/04 F

1. 5.25. Also: (c) Assume that
   
   $$f(x) = e^{-\vert x-2\vert}$$
   
   
   In a single very neat plot, draw $u(x,1)$, $u(x,2)$, and $u(x,3)$ versus $x$. Make sure you draw a complete covering of characteristics in the $x,y$-plane. And show the path of the singularity as a fattened characteristic in the $x,y$-plane.

2. 5.26b. Ignore the hint. Include a very neat sketch of the complete set of characteristic lines. Fatten the asked characteristic in the $x,y$-plane. Simplify your answer as much as possible.

3. 5.27(a). Include a very neat sketch of the complete set of characteristic lines. Is the solution you get valid everywhere?

4. 5.27(b). Do not try to use an initial condition written in terms of two different, related, variables. Get rid of either $x$ or $y$ in the condition. Then call the argument of your undetermined function $\gamma$ and rewrite its expression in terms of $\gamma$. Include a sketch of the complete set of characteristic lines.

5. 5.29 Explain why there is no solution.