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I have problem deriving the solution for cylinder with radial symmetry given:

[tex] \nabla^2U(\rho,z)=R''+\frac{1}{\rho}R'+\frac{Z''}{Z}=0[/tex]

Which give [itex] \rho^2 R''+ \rho R' -k\rho^2 R=0 \hbox { and } Z''+kZ=0[/itex]

With given boundary conditions [itex] U(\rho,0) = U(\rho,h) =0 \hbox { for } \rho<a \hbox{ and } U(a,z) = f(z)[/itex]

The book claimed [itex]k=-\lambda^2 = -ve[/itex] can only produce trivial solution of R. I cannot verify this.

k=-ve give [itex]\rho^2 R''+ \rho R' +\lambda^2 R=0[/itex]

This is parametric Bessel's equation of order zero which give [itex]R(\rho)=c_1 J_0(\lambda_n\rho) + c_2 Y_0(\lambda_n\rho)[/itex]

R(0) is bounded [itex] \Rightarrow c_2 = 0 \Rightarrow R(\rho)= c_1 J_0(\lambda_n\rho)[/itex]

That's where I get stuck. I cannot rule this out [itex] R(\rho)= c_1 J_0(\lambda_n\rho)[/itex] as a solution with the given boundary condition.

The book claimed only k=+ve would give solution. Please give me some suggestion.

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# Question on Dirichlet problem in cylinder with radial symmetry.

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