Higher-order eigenvalue problems

Suppose you have a beam of length 1 with hinged ends. Let $y$ be the transverse deviation of the beam at position $x$ on the beam ($0 < x < 1$). You know that the constants are such that this satisfies the equation $y_{tt} + 4 y_{xxxx} = 0$. Suppose you know that the initial shape of the beam is the graph of $\sin (\pi x)$, and the initial velocity is 0. Solve for $y$.

Suppose you have a beam of length 10 with two fixed ends. Let $y$ be the transverse deviation of the beam at position $x$ on the beam ($0 < x < 10$). You know that the constants are such that this satisfies the equation $y_{tt} + 9 y_{xxxx} = 0$. Suppose you know that the initial shape of the beam is the graph of $\sin^2(\pi x)$, and the initial velocity is equal to $x(10-x)$. Set up the equation together with the boundary and initial conditions. Just s…