Chapter 13.7, Problem 7E

To determine

The solution $u\left(x,y\right)$ of the differential equation $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$ such that ${\frac{\partial u}{\partial x}|}_{x=0}=0\text{, }u\left(1,y\right)=u_0,u\left(x,0\right)=0\text{ and }{\frac{\partial u}{\partial y}|}_{y=1}=0$.