. Question 1.2.9 Consider a thin one-dimensional rod without sources of thermal energy whose lateral surface area is not insulated. 1. Assume that the heat energy flowing out of the lateral sides per unit surface area per unit time is w(x, t). Derive the partial differential equation for the temperature u(x, t). 2. Assume that w(r, t) is proportional to the temperature difference between the rod u(x, t) and a known outside temperature y(x, t). Derive P cp at Ko ax lu(x, t) - Y(x, t) ]h(a) where h(x) is a positive x-dependent proportionality, P is the lateral perimeter, and A is the cross-sectional area. 3. Compare the above equation with the one-dimensional equation for the rod whose lateral surfaces are insulated, but with heat sources