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# voltage across the capacitor

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You really don't have to derive the diff eq for the RLC circuit because we already know its of the form: s^2LC+sRC+1 =0, where v(t) = Ae^(st), dv/dt = sAe^(st) ...

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In the above circuit, you may assume that i(0) = 0 and v(0) = 0
a) suppose L = 20 mH, C = 1 uF, R = 40 ohms, and is = 350 mA. Find v(t) for t >0.

b) suppose L = 0.25 H, C = 12.5 uF, R = 450 ohms, and is = 70 mA. Find v(t) for t >0.

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This solution finds the voltage across the capacitor when t>0 in the RLC circuit. The solution is detailed and has a '5/5' rating.

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I still want to make it clear about how we get the above equations. Also, we need to take care of the current source.
All nodal voltages are labeled in the circuit.
In the s-domain, the capacitor is , the inductor is sL.
The current in the capacitor branch is: (please see the attached file)
Therefore, the voltage of nodal v1 is

Then, the current in the resistor branch is

Apply KCL for nodal v1, we have

Rearrange the equation we have:
(*) (please see the attached file)
Pay attention to the right side of the ...

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