Consider the steady$-$state, two$-$dimensional conduction in a

square rod shown in the figure.

$\frac{d^{2}T}{d{x}^2}+\frac{d^{2}T}{d{y}^2}=0$

Three sides of the rod are insulated, while there is a specified

heat flux at the right surface given by

$\frac{dT}{dx}=\frac{1}{2}-y$

a$)$ Define the boundary conditions.

b$)$ Is there any need for a variable transformation in this problem? Explain your reasoning. If

so$,$ suggest a variable transformation for the dependent variable.

c$)$ Choose the sign of ${}^2$ term. Explain your reasoning.

d$)$ Solve the problem using separation of variables method.