Chapter 13.2, Problem 15E

(a)

To determine

To show: The separated differential equation of the partial differential equation $g\frac{\partial }{\partial x}\left(x\frac{\partial u}{\partial x}\right)=\frac{{\partial }^{2}u}{\partial t^2}$ is $x{X}^{″}+X^{\prime }+\lambda X=0$ and obtain the solution for $x=\frac{{\tau }^2}{4}$.

(b)

To determine

The horizontal displacement $u\left(x,t\right)$ of a chain of length $L$ such that $g\frac{\partial }{\partial x}\left(x\frac{\partial u}{\partial x}\right)=\frac{{\partial }^{2}u}{\partial t^2}$, $u\left(L,t\right)=0,u\left(x,0\right)=f\left(x\right)\text{ and }{\frac{\partial u}{\partial t}|}_{t=0}=0$.