For this problem the best way to separate variables is to considerG(r)φ(θ) and obtain the equationsrGddrrdGdr=-1φd2φdθ2=-λ.Considering theG(r) equation yields the boundary value problemddrrdGdr+λ1rG= 0G(a) =G(b) = 0We now have a regular Sturm-Liouville eigenvalue problem, which is why we chose the separation constantto take the form-λ. The solution to this problem has the same form as in§2.5.2 of Haberman, only nowwe have complex numbers in the exponentG(r) =c1ri√λ+c2r-i√λ=c1ei√λlnr+c2e-i√λlnrwhere the equalityr=elnris used in the second equality. It takes some work, but this latter expressioncan be rewritten asG(r) =c3cos√λlnra+c4sin√λlnra.We now have 0 =G(a) =c3·1 +c4·0 =c3so thatG(r) collapses toG(r) =c4sin√λlnrawhich means0 =G(b) =c4sin√λlnba⇒√λlnba=nπ⇒λ=nπln(b/a)2.Factoring in
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Trigraph, Boundary value problem, dr r, (b/a), dr sinh