A metallic cylinder has radius, R and length, L. All surfaces of the cylinder are kept at a temperature, TS, except for the top surface which is kept at a temperature, T0. Determine the steady-state temperature distribution in the cylinder.

a) State the appropriate boundary conditions and introduce a new variable so that three of the boundary conditions are homogeneous.

b) Solve the PDE using separation of variables and apply the three homogeneous boundary conditions. (bessel functions may be needed)

c) Apply the final boundary conditions and the orthogonality condition. (Note that after separation of variables the ODE is a Sturm-Liouville problem and therefore the resulting eigenfunctions must be orthogonal)