Three sides of a thin rectangular plate or a long rectangular rod are maintained at a constant temperature $T_1,$ while the fourth side is maintained at a constant temperature $T_2.$ We will use the twodimensional, steady$-$state conditions with no heat generation and constant thermal conductivity as following equation and figure.

$\frac{de{l}^{2}T}{del{x}^2}+\frac{de{l}^{2}T}{del{y}^2}=0$

$(1)$ Establish the boundary conditions. $($Use the transformation between $T$ and $.)$

Using the method of separation of variable $($S$.$O$.$V$.),$ derive the full governing equation for temperature profile.