Chapter 8.1, Problem 1E

Prove that the functions (a) $u(x,t)=e^{2t+x}+e^{2t-x}$, (b) $u(x,t)=e^{2t+x}$

are solutions of the heat equation $u_t=2u_{xx}$ with the specified initial boundary conditions:

$\begin{array}{l}\text{a}\text{. }\{\begin{array}{c}u(x,0)=2\cosh x\text{ for }0\le x\le 1\\ u(0,t)=2e^{2t}\text{ for }0\le x\le 1\\ u(1,t)=\left(e^2+1\right)e^{2t-1}\text{ for }0\le x\le 1\end{array}\\ \text{b}\text{. }\{\begin{array}{c}u(x,0)=e^x\text{ for }0\le x\le 1\\ u(0,t)=2e^{2t}\text{ for }0\le t\le 1\\ u(1,t)=e^{2t+1}\text{ for }0\le t\le 1\end{array}\end{array}$