# Overdamped RLC circuit

I have a series RLC circuit with an equation:

$$\frac{d^2I}{dt} + 2\alpha \frac{dI}{dt} + \omega_0^2 I = 0$$

(No outside sources affecting the circuit, only some $$I_0$$ was in the circuit at the beginning)

The result is, for overdamped response ($$\alpha^2 - \omega_0^2 > 0$$):

$$I = Ae^{(-\alpha - \sqrt{\alpha^2 - \omega_0^2})t} + Be^{(-\alpha - \sqrt{\alpha^2 - \omega_0^2})t}$$

So, if $$I(0) = I_0$$ then $$A+B = I_0$$, but that still doesn't solve A and B completely. What other initial conditions do i have to include or what other things do i have to do in order to find A and B completely?

As in any other second-order ODE, the two initial conditions needed are $$I(0)$$ and $$\frac{dI}{dt}(0)$$.
• @lkky7 $dI/dt(0) = 0$ gives you an equation containing your constants $A$ and $B$, so you only have one arbitrary constant not two. Oct 27, 2018 at 22:16