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Exercise 7: A consideration of practical application leads to Laplace's equation, which reduces to an ODE.

Consider the steady state temperature in a circular gun barrel. Assuming radial symmetry, and no variation in temperature along the length of the barrel, leads to Laplace's equation for temperature as a function of the radial distance r from the center of the barrel. With temperature given by for , the Laplace PDE reduces to the ODE .