Find the eigen values and eigen vectros for a rod.

1. If a rod is free at both ends.
2. If a rod is fixed at both ends.

if the rod is free at both ends:

au/ax(0,t)=0

au/ax(l,t)=0

sinwl/c=0 (caracteristic eqn.)

wj= j*pi*c/l=(j*pi/l)*(E*) j= 1,2,3,

j= A_2j* cos(w_jx/c)

if the rod is fixed at both ends:

u(0,t)=0

u(l,t)=0

sin(wl/c)=0

wj= j*pi*c/l=(j*pi/l)*(E*) j= 1,2,3,

j= A_1j* sin(j*pi*x/l) j= 1,2,3,