Section 1.4: 1, 3, 4, 5
Terminology: "steady state" and "equilibrium solution" mean the same thing, namely that the solution depends only on position and not on time. In other words, u=u(x) is independent of t. (Hence u_t=0.)
(#3: note that D is a region in 3 dimensions, and x=(x₁,x₂,x₃) is a point in D. In this problem, you can assume on physical grounds that the steady state is a constant, u(x)=C. Your task is to find C in terms of the given information.)
(#4: the answer in the back of the book is off by a minus sign)

Section 2.1: 1, 5, 7, 8, 9
(#5: in the Hint, you are taking the intersection of two intervals and then finding the length of that intersection - why does this evaluate the given integral?)
(#7: recall a function is odd if h(-x) = -h(x) for all x.)

Section 2.2: 1, 2, 3, 5

Section 2.2:
Consider the wave equation in a bounded region D in three dimensions, with homogeneous Dirichlet boundary conditions.
Question 1. Show that energy is conserved. (Here we use the definition: energy=(1/2) times the integral over D of u_t² + c² |gradient u|².) (Hint: |gradient u|²=gradient u dot gradient u. Another hint: divergence theorem.)
Question 2. Is energy also conserved under homogeneous Neumann boundary conditions?
Question 3. State and prove an energy dissipation result for the diffusion equation in three dimensions, under homogeneous Dirichlet or Neumann boundary conditions.

Section 2.3:
3, 6, 8
(#3(a): Just show u(x,t) is greater than or equal to zero. #3(c): I encourage you to use Iode. Play around with the "Plot resolution" option, in order to get a reasonably accurate graph.)

Section 2.4:
1

Section 5.5: 10
Remarks: You are asked to prove Theorem 5.4.2 (as stated in class, not in the book) for the case of sine series and cosine series. But we have proved it already for the full Fourier series, in Section 5.5, and so you may use that result here to help you.

Section 5.6: 6, 13

Section 6.1: 4, 5, 2, 3 (but in problems 2 and 3, change k² to -k²; this changes the answers of course!)
Hint for #3: see page 252 for Bessel's equation and functions

Additional problem:
Show by example that term-by-term differentiation of the Fourier series of a function f(x) can fail to give the series of the derivative df/dx, for a function f that jumps. You could consider the sine series of f(x)=1, for example. (Aside: this sort of problem is why the Fourier series of u_xx does not just equal the second x-derivative of the Fourier series for u, in Section 5.6.)