Second order ODE for RLC circuit

[icesalmon]

if I consider a circuit consisting of a capacitor, an inductor and a resistor and using kirchhoffs voltage rule for the circuit i come up with the following

L(Q''(t)) + R(Q'(t)) + (Q(t))/C = 0 I solve for the roots using a characteristic equation of the form
LM² +MR +(1/C) = 0
solving this for m I obtain
m = [-R/L +/- sqrt((r/l²) - 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^-Btcos(w't+θ)
B = R/2L

 

[zoki85]

See:
https://www.physicsforums.com/threads/second-order-linear-de.286529/#post-2044152

 

[icesalmon]

so obviously the characteristic equation doesn't directly lead to that damped charge motion then, it depends on the discriminant. I thought I was SUPPOSED to get the decaying exponential out of it. Anyways, thanks.