Math 641-600 Suggested Problems (Fall 2010)

Suggested Problems for Chapter 4 - These are not to be handed in.

• Section 4.1: 5, 8
• Section 4.2: 2, 4
• Section 4.3: 2, 3, 5
• Let Lu = −(xu′ )′, D(L) := {u, Lu ∈ L²[1,2], u(1) = u(2) = 0}.
  1. Show that L is self adjoint and positive definite with respect to the inner product for L²[1,2].
  2. Find the Green's function for L.
  3. Show directly — i.e., without quoting Theorem 4.7 — by using the Green's function that the eigenfunctions of L are complete in L²[1,2].

Updated 12/5/10.