Problem $1.$ Consider a room with a double$-$pane window for insulation and protection. Assume that all the insulation layers have the same cross$-$sectional area $A,$ the thermal conductivity $ki$ and thickness $i$ of the $i-$th layer. A heat flow rate qin is applied to the room with capacitance $C.$ On the right side of the third layer, convective heat transfer takes place, with $h$ being the convective coefficient, $Ts$ the surface temperature, and $Tf$ the temperature of a cold air.

Derive the differential equation of the system in terms of the room temperature $T$ and determine the transfer function from $q$ in to $T$ and the transfer function from $Tf$ to $T.$