2 a02 1 a03 2 a04 1 a11 3 b11 2 vibmem

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Unformatted text preview: , the left-hand side does not depend on θ while the right-hand side does not depend on r. Thus neither side can depend on either θ or r, and hence both sides must be constant: − rd R dr r dR dr 1 d2 Θ = µ. Θ dθ2 + λr2 = Thus in the manner now familiar, the partial differential equation divides into two ordinary differential equations d2 Θ = µΘ, dθ2 r d dr r dR dr Θ(θ + 2π ) = Θ(θ), − λr2 R = −µR, R(a) = 0. Once again, the only nontrivial solutions are µ = 0, Θ= a0 2 and µ = −n2 , Θ = an cos nθ + bn sin nθ, for n a positive integer. Substitution into the second equation yields r Let x = √ d dr r dR dr + (−λr2 − n2 )R = 0. −λr so that x2 = −λr2 . Then since r d d =x , dr dx our differential equation becomes x d dx x where R vanishes when x = x d dx x dy dx √ dR dx + (x2 − n2 )R = 0, (5.29) −λa. If we replace R by y , this becomes + (x2 − n2 )y = 0, 145 √ y ( −λa) = 0. (5.30) The differential equation appearing in (5.30) is Bessel’s equation, which we studied in Section 1.4. Recall that in Section 1.4, we...
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This document was uploaded on 01/12/2014.

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