The displacement for the transverse vibrations of a beam satisfies the differential equation

utt + (b^2)uxxxx = 0

Find u(x,t) if u(0,t) = 0, u(L,t) = 0, uxx(0,t) = 0, uxx(L,t) = 0, u(x,0) = f(x), ut(x,0) = g(x)

I've never encountered a differential equation like this before. How do you solve it?