Chapter 10.5, Problem 13E

Find a formal solution to the initial boundary value problem

$\frac{\partial u}{\partial t}=2\frac{{\partial }^{2}u}{\partial x^2}+4x$, $0<x<\pi$, $t>0$,

$\begin{array}{rll}u\left(0,t\right)& =& u\left(\pi ,t\right)\\ & =& 0\end{array}$, $t>0$,

$u\left(x,0\right)=\sin x$, $0<x<\pi$.