7 02/25 F

1. 2.19b, h. Show a picture of the different regions.

2. Notes 1.3.2.1

3. Notes 1.4.3.1

4. Notes 1.4.4.1

5. 2.22b,g. Draw the characteristics very neatly in the $xy$-plane,

6. 2.28d. First find a particular solution. Next convert the remaining homogeneous problem to characteristic coordinates. Show that the homogeneous solution satisfies
   
   $$2 u_{h,\xi\eta} = u_{h,\eta}$$
   
   
   Solve this ODE to find $u_{h,\eta}$, then integrate $u_{h,\eta}$ with respect to $\eta$ to find $u_h$ and $u$. Write the total solution in terms of $x$ and $y$.

7. 2.28f. In this case, leave the inhomogeneous term in there, don't try to find a particular solution for the original PDE. Transform the full problem to characteristic coordinates. Show that the solution satisfies
   
   $$4 u_{\xi\eta} - 2 u_{\xi} \pm e^\eta = 0$$
   
   
   where $\pm$ indicates the sign of $xy$, or
   
   $$4 \xi\eta u_{\xi\eta} - 2\xi u_{\xi} + \eta = 0$$
   
   
   or
   
   $$4 \xi\eta u_{\xi\eta} + 2\xi u_{\xi} - \frac{1}{\eta} = 0$$
   
   
   or equivalent, depending on exactly how you define the characteristic coordinates. Solve this ODE for $u_{\xi}$, then integrate with respect to $\xi$ to find $u$. Write the solution in terms of $x$ and $y$.