I have included a Wave Equation problem with parts a-c, that has variable tension. It involves separation of variables, the Sturm-Liouville system, and an application to the "Rayleigh Quotient" involving the Eigenvalues. I have included notes on the Sturm-Liouville system with examples and properties. Please refer to these notes to maintain continuity for the solution needed. Thank you for your time and consideration in these matters.
1.) Given the wave equation below (with variable tension τ(x) = x) and appropriate Boundary Conditions:
(a) Apply Separation of Variables to the PDE to get 2 ODE's.
(b) Show that the Eigenvalue problem is a singular Sturm-Liouville system; identify the functions p(x), w(x), and q(x). Do not try to solve it.
(c) Based on the equation below, called the "Rayleigh quotient", do you expect this problem to have any negative Eigenvalues? How about a zero Eigenvalue? Be sure to use specific p(x), w(x), q(x) and Boundary Conditions for this problem.
This provides an example of solving a wave equation problem with variable tension using separation of variables and Rayleigh quotient.