Chapter 8.3, Problem 3E

Prove that the functions (a) $u(x,y)=e^{-\pi y}\sin \pi x$, (b) $u(x,y)=\sinh \pi x\sin \pi y$ are solutions of the Laplace equation with the specified boundary conditions:

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