Chapter 10.6, Problem 1E

In Problems 1 -4 , find a formal solution to the vibrating string problem governed by the given initial-boundary value problem.

$\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}$, $0<x<1$, $t>0$,

$u\left(0,t\right)=u\left(1,t\right)=0$, $t>0$,

$u\left(x,0\right)=x\left(1-x\right)$, $0<x<1$,

$\frac{\partial u}{\partial t}\left(x,0\right)=\sin 7\pi x$, $0<x<1$