6 02/20 F

1. 2.23. Reduce to canonical form by rotating the coordinate system. (Not using characteristic coordinates as the book does.) What is the angle the coordinate system must be rotated over?

2. 2.24.

3. By identifying $d'$ show that the PDE of the previous question may be reduced to
   
   $$u_{\xi_1\xi_1} + 3 u_{\xi_2\xi_2} + 4 u_{\xi_3\xi_3} \pm \... ...8}{\sqrt{2}} u_{\xi_2} \pm \frac{20}{\sqrt{3}} u_{\xi_3} = 0$$
   
   
   where the $\pm$ depend on how you choose the sign of your eigenvectors. Multiply the above equation by 6 and then rescale the independent variables to get
   
   $$u_{\xi\xi} + u_{\eta\eta} + u_{\theta\theta} \pm 16 u_{\xi} \pm 8 u_{\eta} \pm 10\sqrt{2} u_{\theta} = 0$$
   
   

4. Get rid of the first derivatives in the previous equation by defining a new unknown $v=u/e^{\alpha\xi+\beta\eta+\gamma\theta}$ where $\alpha$, $\beta$, and $\gamma$ are constants to be found from the condition that the first order derivatives disappear. Write and name the final equation.

5. 2.22b,g. Draw the characteristics 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 for $u_{h,\eta}$, then integrate with respect to $\eta$ to find $u_h$ and $u$. Write the 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 $y$.