Consider the two$-$dimensional steady$-$state temperature distribution on a rectangular metal plate with dimensions of $\backslash (2$ m $\backslash$times $1$ m $\backslash ),$ shown in the figure below, where the top boundary is maintained at a temperature of $\backslash (10^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ while three other boundaries are maintained at $\backslash (0^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The metal plate has no heat source.

a$.$ Specify the partial differential equation and the boundary conditions that define the problem.

b$.$ Using the separation$-$of$-$variables technique, find the analytical expression for the temperature distribution, $\backslash ($ T$($x$,$ y$)\backslash ),$ in the metal plate.