if I consider a circuit consisting of a capacitor, an inductor and a resistor and using kirchoffs voltage rule for the circuit i come up with the following(adsbygoogle = window.adsbygoogle || []).push({});

L(Q''(t)) + R(Q'(t)) + (Q(t))/C = 0 I solve for the roots using a characteristic equation of the form

LM^{2}+MR +(1/C) = 0

solving this for m I obtain

m = [-R/L +/- sqrt((r/l^{2}) - 4(1)(1/LC))]/2

i'm expecting an equation using both decaying exponentials and sinusoids so that the energy will tend towards zero after a long time. But this depends on the values of R, C and L. I'm having trouble moving forward from this point in deriving the equation for the charge on the capacitor as a function of time.

Q(t) = Ae^{-Bt}cos(w't+θ)

B = R/2L

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# Second order ODE for RLC circuit

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