Boundary value problems

Consider a spinning string of length 2 and linear density 0.1 and tension 3. Find the smallest angular velocity when the string pops out.

Suppose $x'' + \lambda x = 0$ and $x(0)=1$, $x(1) = 1$. Find all $\lambda$ for which there is more than one solution. Also find the corresponding solutions (only for the eigenvalues).

Suppose $x'' + x = 0$ and $x(0)=0$, $x'(\pi) = 1$. Find all the solution(s) if any exist.

Consider $x' + \lambda x = 0$ and $x(0)=0$, $x(1) = 0$. Why does it not have any eigenvalues? Why does every first-order equation with two endpoint conditions such as the one above have no eigenvalues?

Suppose $x''' + \lambda x = 0$ and $x(0)=0$, $x'(0) = 0$, $x(1) = 0$. Suppose that $\lambda > 0$. Find an equation that all such eigenvalues must satisfy. Hint: Note that $-\sqrt[3]{\lambda}$ is a root of $r^3+\lambda = 0$.