the governing differential equation of the temperature excess is recognized as per the Euler equation
The result you are looking for can be found in various heat transfer textbooks. One source that you can look at is using GoogleBooks. Check out the following link: http://tinyurl.com/gtzqbe3
If you are interested how one derives the axisymmetric fin equation for arbitrary geometry and its solution using Mathematica you can take a look at the attached file. In this case the cross section of the fin varies with axial position, thus the BVP has variable coefficients and thus cannot be solved analytically.
If the fin is a "body of revolution" of a parabola, the fin efficiency is 2/((4/9(mL)**2+1)**0.5+1) where m is (4h/(kD))**0.5, D is the diameter of circular fin base, L is its height, and k and h are thermal conductivity and convection coefficient. If the fin is instead a body of extension of a parabola, the efficiency is 2/((4(mL)**2+1)**0.5+1) where m is (2h/(kt))**0.5 and t is the thickness of the fin base. These results are from the Incropera et al text "Fundamentals of Heat and Mass Transfer", but are also available in many other books.
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