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A metallic spherical shell occupying the region given in terms of spherical polar coordinates \( (r, \theta, \phi)\) by \(r \le a \) has its surface \(r = a\) maintained at temperature

\(u(a,\theta)=1 + \cos(\theta)+2 \cos^2(\theta) \)

Using the general solution for the Laplace equation \( u(r,\theta)=\sum A_n r^n P_n \) where the Pn are legendre polynomials, find the (axisymmetric) steady state temperature distribution \(u(r,\theta) \) within the shell.<<there was a long incomprehensible expression here which seems to have disappeared ?>>

You may assume the legrendre polynomials (first three) and that Legendre polynomials satisfy the orthogonality relation.

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