Chapter 10.6, Problem 9E

If one end of a string is held fixed while the other is free, then the motion of the string is governed by the initial-boundary value problem

$\frac{{\partial }^{2}u}{\partial t^2}={\alpha }^2\frac{{\partial }^{2}u}{\partial x^2}$, $0<x<L$, $t>0$

$u\left(0,t\right)=0$ and $\frac{\partial u}{\partial x}\left(L,t\right)=0$, $t>0$

$u\left(x,0\right)=f\left(x\right)$, $0<x<L$

$\frac{\partial u}{\partial t}\left(x,0\right)=g\left(x\right)$, $0<x<L$

Derive a formula for a formal solution.