6 02/19 F

1. 2.20.

2. 2.21, in three spatial dimensions, and time as appropriate.

3. 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?

4. 2.24.

5. 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 signs you will end up with depend on how you choose the sign of your eigenvectors. Multiply the equation you got by 6 and then rescale the independent variables to get an equation of the form
   
   $$u_{\xi\xi} + u_{\eta\eta} + u_{\theta\theta} \pm 16 u_{\xi} \pm 8 u_{\eta} \pm 10\sqrt{2} u_{\theta} = 0$$
   
   

6. Get rid of the first derivatives in the obtained 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.

7. 4.20. A number $T$ is rational if it can be written as the ratio of a pair of integers, e.g. 1.5 = 3/2 = 6/4 = 9/6 = .... It is irrational if it cannot, like $\sqrt{2}$. Near any rational number, irrational numbers can be found infinitely closely nearby, and vice versa. For example, the value $\pi$ to one billion digits, as found on the internet, is the rational number $31415927\ldots/10000000\ldots$; $\pi$ itself is not rational. The wave equation problem when $T=\pi$ has no nonzero solutions, but when $T=\pi$ to 1 billion digits has infinitely many of them. Obviously, in physics it is impossible to determine the final time to infinitely many digits, so there is no physically meaningful solution to the stated problem.

   For nonzero solutions, try $u=\sin(n\pi x)\sin(n\pi t)$. Show that this satisfies the wave equation and the boundary conditions at $x=0$ and $x=1$ and the initial condition at $t=0$. See when it satisfies the end condition at $t=T$.

   The wave equation needs two initial conditions at $t=0$, not one condition at $t=0$ and one at $t=T$.

8. Show that the Laplace equation
   
   $$u_{xx} + u_{yy} = 0 \qquad 0 \le x \le 1 \quad 0 \le y \le T$$
   
   
   with the same boundary conditions, (replacing $t$ by $y$), does not have the same problem. Hint: assume solutions of the form $u=\sin(n\pi x)f(y)$ and plug into the Laplace equation and $u(x,0)=0$ to figure out what $f(y)$ is. (You may want to recall the graphs of the hyperbolic functions.)