Consider the welding situation of the previous problem. This time let's consider the case where we have convective heat transfer from the rod's surface. For a rod of radius, \(r_{o}\), and a heat transfer coefficient and free stream temperature of \(h\) and \(T_{\infty}\), respectively:

a. Derive the differential equation governing the temperature profile in the rod. Assume it acts as a fin with no radial temperature gradients.

b. Solve the equation using the same boundary conditions as in the previous problem.

c. Reconsider the problem if the rod also experiences resistance heating with a resistance, \(\mathcal{R}_{e}\).