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LC natural response derivation 4

In this final step of the derivation, we find two initial conditions and use them to come up with a sinusoidal solution for the LC natural response. Created by Willy McAllister.

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• Is this another case of solving a problem in the set of real numbers through complex numbers or the complex solution is related to another dimension of the phenomenon?

I'm thinking, for instance, about electromagnetic waves with a couple of orthogonal fields, electric and magnetic. In this case the real solution and the imaginary solution could describe this situation. In our LC model we have an equilibrium between a magnetic field, (inductor), and an electric field (capacitor).

Nature is amazing, but mathematics is astonishing!
• I lean towards your first explanation. Imaginary numbers appear in the middle of a journey to a real solution. Once we have the abstract i-v equations of L and C with their derivative terms, the physical phenomena that gave rise to those equations is left behind. The appearance of the imaginary is self-contained within the math.