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I have attempted two methods:

1) Basically take a homogeneous second-order differential for any spring, and equate it to a "driving" force equal to the force between two charged plates as a function of the distance between them;

2) And to take the homogeneous second order differential for any LC circuit, and equate it to the charge on any capacitor as a function of the distance between the plates.

My problem in case one is that the distance between plates is in fact equal to the amplitude of the spring's motion minus its actual position, quantity squared; and I simply have no idea where to begin solving that. The second attempt certainly looks less complex, but the distance between plates in that case is in fact equal to the entire solution to a homogeneous spring equation, which just makes it as complex as case one.

As for the resonant frequencies, my intuition wants to say that if the circuit and spring w's are equal that should represent total system resonance; but then, that seems too simple considering the differential models I just delineated. Any insights?