Chapter 8.1, Problem 2E

Prove that the functions (a) $u(x,t)=e^{-\pi t}\sin \pi x$, (b) $u(x,t)=e^{-\pi t}\cos \pi x$ are solutions of the heat equation $\pi u_t=u_{xx}$, with the specified initial boundary conditions:

$\begin{array}{l}\text{a}\text{. }\{\begin{array}{c}u(x,0)=\sin x\text{ for }0\le x\le 1\\ u(0,t)=0\text{ for }0\le t\le 1\\ u(1,t)=0\text{ for }0\le t\le 1\end{array}\\ \text{b}\text{. }\{\begin{array}{c}u(x,0)=\cos \pi x\text{ for all }0\le x\le 1\\ u(0,t)=e^{-\pi t}\text{ for }0\le t\le 1\\ u(1,t)=-e^{-\pi t}\text{ for }0\le t\le 1\end{array}\end{array}$