2) If the rod is fastened at one end, x = 0, and free at the...

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2) If the rod is fastened at one end, x = 0, and free at the other, x = L, we have u(0,t) = 0 and ux (L,t) = 0. Show that the motion corresponding to initial displacement u(x, 0) = f(x) and initial velocity zero is -N= u = An sin pnx cos pnct, where An Ln=0 2 = f(x) sin prx dx, and P = (2n+1) 2L 2) If the rod is fastened at one end, x = 0, and free at the other, x = L, we have u(0,t) = 0 and ux (L,t) = 0. Show that the motion corresponding to initial displacement u(x, 0) = f(x) and initial velocity zero is -N= u = An sin pnx cos pnct, where An Ln=0 2 = f(x) sin prx dx, and P = (2n+1) 2L