Exercise 105

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 $r^3+\lambda = 0$.