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Unformatted text preview: , the lefthand side does not depend on θ while
the righthand 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.
 Winter '14
 Equations

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