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I have been given the following problem as an assignment.

[tex] \frac{\partial ^2 \phi}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial \phi}{\partial \rho} + \frac{1}{\rho^2}\frac{\partial \phi}{\partial \chi^2} + \frac{\partial ^2 \phi}{\partial Z^2}+B^2\phi = 0 [/tex]

Here is my attempt to the problem:

Assuming [tex] \phi = S(\rho,\chi)Z(z) [/tex]

[tex] \frac{1}{S(\rho,\chi)}\frac{\partial ^2 S(\rho,\chi)}{\partial \rho^2} + \frac{1}{S(\rho,\chi) \rho }\frac{\partial S(\rho,\chi)}{\partial \rho} + \frac{1}{S(\rho,\chi) \rho^2}\frac{\partial S(\rho,\chi)}{\partial \chi^2} + \frac {1}{Z}\frac{\partial ^2 Z}{\partial Z^2} + B^2 = 0 [/tex]

Separating the variables we get

[tex] \frac{\partial ^2 z}{\partial Z^2} + B^2 Z = 0 [/tex]

[tex] \frac{1}{S(\rho,\chi)}\frac{\partial ^2 S(\rho,\chi)}{\partial \rho^2} + \frac{1}{S(\rho,\chi) \rho }\frac{\partial S(\rho,\chi)}{\partial \rho} + \frac{1}{S(\rho,\chi) \rho^2}\frac{\partial S(\rho,\chi)}{\partial \chi^2} + B^2 = 0 [/tex]

Assuming [tex] S(\rho, \chi) = \rho(\rho) \chi(\chi) [/tex]

[tex] \frac{1}{\rho}\frac{\partial ^2 \rho}{\partial \rho^2} + \frac{1}{\rho^2 }\frac{\partial \rho}{\partial \rho} + \frac{1}{\rho^2 \chi}\frac{\partial ^2 \chi}{\partial \chi^2} + B^2 = 0 [/tex]

How can I solve this last PDE?

Thank you in advance

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# Homework Help: Partial Differentiation Problem

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