Chapter 5.2, Problem 32E

(a)

To determine

To show: The Eigenvalues ${\lambda }_n={\alpha }_n^4$ for the differential equation $EI\frac{d^{4}y}{d{x}^4}-\rho {\omega }^{2}y=0$ are defined by the positive roots of $\cos \left(\alpha \right)\cos \text{h}\left(\alpha \right)=1$.

(b)

To determine

To show: The Eigen functions for the differential equation $EI\frac{d^{4}y}{d{x}^4}-\rho {\omega }^{2}y=0$ are $y_n\left(x\right)=\left(-\sin \left({\alpha }_n\right)+\sin \text{h}\left({\alpha }_n\right)\right)\left(\cos \left(x{\alpha }_n\right)-\cos \text{h}\left(x{\alpha }_n\right)\right)+\left(\cos \left({\alpha }_n\right)-\cos \text{h}\left({\alpha }_n\right)\right)\left(\sin \left(x{\alpha }_n\right)-\sin \text{h}\left(x{\alpha }_n\right)\right)$ where, $n=1,2,3\dots$.