= Overdamped RLC response
{title2=$\alpha>\omega_0$}
An <overdamped RLC response> is a free <series RLC circuit> response with $R>2\sqrt{L/C}$, equivalently $\alpha>\omega_0$. Put $\beta=\sqrt{\alpha^2-\omega_0^2}$. The distinct negative <characteristic roots> are $r_\pm=-\alpha\pm\beta$, so the general <electric charge> is
$$
q(t)=Ae^{r_+t}+Be^{r_-t},\qquad i(t)=q'(t).
$$
For <initial conditions> $q(0)=q_0$, $i(0)=0$,
$$
q(t)=q_0e^{-\alpha t}\left(\cosh(\beta t)+\frac\alpha\beta\sinh(\beta t)\right),\qquad
i(t)=-\frac{q_0\omega_0^2}{\beta}e^{-\alpha t}\sinh(\beta t).
$$
For $q_0>0$, this <electric charge> decreases to zero without reversing sign. In the dimensionless example $\omega_0=1$, $\alpha=2$, the <characteristic roots> are $-2\pm\sqrt3\approx-0.268,-3.732$. The slow term dominates late times. Stronger <damping> can therefore make the final return slower: for $\alpha\gg\omega_0$, $r_+\sim-\omega_0^2/(2\alpha)=-1/(RC)$. Nonoscillation alone does not imply a fast response, and monotonicity here uses the specified <initial conditions>.
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