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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?

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1 Answer 1

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As in any other second-order ODE, the two initial conditions needed are $I(0)$ and $\frac{dI}{dt}(0)$.

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  • $\begingroup$ And if dI/dt(0) = 0, then how do I solve it? $\endgroup$
    – lkky7
    Oct 27, 2018 at 17:48
  • $\begingroup$ @lkky7 how about you take dI/dt at t=0 and see what happens? $\endgroup$ Oct 27, 2018 at 19:36
  • $\begingroup$ @lkky7 $dI/dt(0) = 0$ gives you an equation containing your constants $A$ and $B$, so you only have one arbitrary constant not two. $\endgroup$
    – alephzero
    Oct 27, 2018 at 22:16

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