Problem: Consider torsional oscillations of a homogeneous cylindrical shaft. If w(r, t) is the angular displacement at time t of the cross section at r, then 20 w 01 where the initial conditions are 0xL,t>0 -(,0)=0, 0xL, w(x,0) = f(x), and and the ends of the shaft are fixed elastically: dw (0,t)-aw(0,1)=0, and (L.t)-aw(L,t)=0 t>0 r where a is a constant. a) Why is it possible to use separation of variables to solve this problem? b) Use separation of variables and show that one of the resulting problems is a regular Sturm-Liouville problem. c) Show that all of the eigenvalues of this regular Sturm-Liouville problem are positive.