Chapter 8.3, Problem 4E

Prove that the functions (a) $u(x,y)=e^{-xy}$, (b) $u(x,y)={\left(x^2+y^2\right)}^{3/2}$ are solutions of the specified Poisson equation with the given boundary conditions:

$\begin{array}{l}\text{a}\text{. }\{\begin{array}{c}\Delta u=e^{-xy}\left(x^2+y^2\right)\\ u(x,0)=1\text{ for 0}\le x\le 1\\ u(x,1)=e^{-x}\text{ for 0}\le x\le 1\\ \begin{array}{l}u(1,y)=1\text{ for 0}\le y\le 1\\ u(1,y)=e^{-y}\text{ for 0}\le y\le 1\end{array}\end{array}\\ \text{b}\text{. }\{\begin{array}{c}\Delta u=9\sqrt{x^2+y^2}\\ u(x,0)=x^3\text{ for 0}\le x\le 1\\ u(x,1)={\left(1+x^2\right)}^{\text{3/2}}\text{ for 0}\le x\le 1\\ \begin{array}{l}u(0,y)=y^3\text{ for 0}\le y\le 1\\ u(1,y)={\left(1+y^2\right)}^{\text{3/2}}\text{ for 0}\le y\le 1\end{array}\end{array}\end{array}$