According to my book, this is a way to decribe an vibrating elastic "beam" (I guess that means "bar".) I have to find its vibrational frequencies, but I don't know really know what that means. I will try to solve it though.
If I first try to find separable solutions, because I thought that you would and up with cosines and sines, rand with some requirement about what should be inside those functions, which would be the frequency I guess. So I took $u(t,x) \ = \ v(t)w(x)$, so that the equation becomes: $$ v''(t)u(x)\ = \ u_{tt} \ = \ -u_{xxxx} \ = \ -v(t)w''''(x) $$ Now we see that $$ \frac{v''}{v} \ = \ -\frac{w''''}{w} \ =: \ \lambda $$ If we distinguish the cases $\lambda > 0, \ \lambda=0, \ \lambda <0$ we can deduce things about the shape of $v$ and $w$. But I'm afraid that the work becomes very cumbersome that the forth derivative of $w$. Can you please help me?
