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417_problem_2_5_9b

# 417_problem_2_5_9b - For this problem the best way to...

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For this problem the best way to separate variables is to consider G ( r ) φ ( θ ) and obtain the equations r G d dr r dG dr = - 1 φ d 2 φ 2 = - λ. Considering the G ( r ) equation yields the boundary value problem d dr r dG dr + λ 1 r G = 0 G ( a ) = G ( b ) = 0 We now have a regular Sturm-Liouville eigenvalue problem, which is why we chose the separation constant to take the form - λ . The solution to this problem has the same form as in § 2 . 5 . 2 of Haberman, only now we have complex numbers in the exponent G ( r ) = c 1 r i λ + c 2 r - i λ = c 1 e i λ ln r + c 2 e - i λ ln r where the equality r = e ln r is used in the second equality. It takes some work, but this latter expression can be rewritten as G ( r ) = c 3 cos λ ln r a + c 4 sin λ ln r a . We now have 0 = G ( a ) = c 3 · 1 + c 4 · 0 = c 3 so that G ( r ) collapses to G ( r ) = c 4 sin λ ln r a which means 0 = G ( b ) = c 4 sin λ ln b a λ ln b a = λ = ln( b/a ) 2 . Factoring in
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