Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to the Brown and Churchill book, the required text for the course. Unless otherwise indicated, all parts of a problem (a), (b), etc. should be completed. You should write up solutions neatly to all problems, making sure to SHOW ALL YOUR WORK. Since many of the problems have the answers included, your work is what's important!
A nonempty subset of the assignment will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be YOUR OWN WORK.
Important: Please list the names of any students or faculty you worked with on the assignment.
Chapter 4, pp. 98 - 99
Chapter 4, pp. 102
Note: For problem #2, follow the same arguments used in class when we solved the same ODE with Neumann boundary conditions. You should do all four cases (zero, positive, negative and complex eigenvalues.)
Chapter 5, pp. 108 - 109
Problem: 1, 2
Note: For problem #1, you are asked to solve the 1d Heat Equation with homogeneous Dirichlet boundary conditions and initial temperature constant at u_0. Use the separation of variables technique to solve the problem completely (including finding the correct Fourier coefficients). Then calculate the flux -K u_x and explain why -K u_x(Pi/2,t) = 0 for all t.
Chapter 5, pp. 113 - 114
Problems: 2, 4
Note: For problem #2, to find the approximate temperatures, you will need to use Maple or a good calculator to approximate the sum of an infinite series. Give the requested temperatures to the nearest 1/100th degree. Note also that t is in seconds not minutes.
Additional Problem: Find all the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville problem X''(x) + lambda X(x) = 0, X(0) = 0, X'(c) = 0 . You may assume that the eigenvalues are all real. This is a mixed boundary condition, with Dirichlet at the left end and Neumann at the right.