Chapter 8.3, Problem 6E

Prove that the functions (a) $u(x,y)=e^{x+2y}$, (b) $u(x,y)=y/x$ are solutions of the specified elliptic equation with the given Dirichlet boundary conditions:

$\begin{array}{l}\text{a}\text{. }\{\begin{array}{c}\Delta u=5u\\ u(x,0)=e^x\text{ for 0}\le x\le 1\\ u(x,1)=e^{x+2}\text{ for 0}\le x\le 1\\ \begin{array}{l}u(0,y)=e^{2y}\text{ for 0}\le y\le 1\\ u(1,y)=e^{2y+1}\text{ for 0}\le y\le 1\end{array}\end{array}\\ \text{b}\text{. }\{\begin{array}{c}\Delta u=\frac{2u}{x^2}\\ u(x,0)=0\text{ for 1}\le x\le 2\\ u(x,1)=1/x\text{ for 0}\le x\le 2\\ \begin{array}{l}u(1,y)=y\text{ for 0}\le y\le 1\\ u(2,y)=y/2\text{ for 0}\le y\le 1\end{array}\end{array}\end{array}$