hey pf!(adsbygoogle = window.adsbygoogle || []).push({});

i was wondering if you could help me out with a pde, namely $$\alpha ( \frac{z}{r} \frac{\partial f}{\partial r} + \frac{\partial f}{\partial z} ) = \frac{2}{r} \frac{\partial f}{\partial r} + \frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial z^2} + 2 \frac{z}{r} \frac{\partial^2 f}{\partial r \partial z}$$

i won't list the boundary conditions, as i'm just trying to find a general solution for now. i tried the substitution ##r^2 = z^2 + y^2## which changed the equation to separable (it's not currently separable, i've tried). i then used a bessel function and an exponential but could not fit them to the boundary conditions. i know an analytical solution exists, but i'm not sure how to get there. the solution seems to be separable in ##r## and ##z## but the equation is not. this is ironic because if i turn the equation into a separable PDE the solution is not separable.

please help me out!

thanks!

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# Tough pde

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