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Here is how: And the expression is (from wikipedia): $$2\pi \varepsilon a\sum_{n=1}^{\infty }\frac{\sinh \left( \ln \left( D+\sqrt{D^2-1}\right) \right) }{\sinh \left( n\ln \left( D+\sqrt{ D^2-1}\right) \right) }$$

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I think that the arrangement looks something like this?

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So the spectral approach as I remember is you make an anszats $i=\sum_k a_k f_k(t)$ or something like that, you use some theorem to say that each component of the sum is independent so you get $a_k f_k''(t) + 5 a_k f_k'(t) + 1/4 a_k f_k(t)=0$, to get the factors (characteristic polynomial is what you said I believe) My guesses where the error could be: I ...

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A capacitor can store charge so if both balls were initially uncharged one could move charge from one ball to the other. Thus the balls now store charge and energy - they can be classed as a capacitor. As the separation between the balls decreases the capacitance of the system increases.

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