Initial conditions for rlc series natural response

[eng_stud]

Homework Statement

Find v(t) across a cap. in a series rlc circuit with no driving force (initial v across cap: 24V)

Homework Equations

from the values of the components, \alpha > \omega_0, the circuit is overdamped, and the following equation can be used: v(t) =A_1 e^{s_1 t} + A_2 e^{s_2 t}

The Attempt at a Solution

My trouble is basically finding another initial condition to solve the 2nd order diff. equation above. At t=0, the voltage across the capacitor is 24V, so: 24 =A_1 + A_2.
The other initial condition I would think should come from the fact that the current in the inductor can not change at once, so initial current is i=0. I'm just not quite sure how to use this. Can I say that, since current in cap: i=C dv/dt, then: i/C = dv/dt = 0 = \frac{d(A_1 e^{s_1 t} + A_2 e^{s_2 t})}{dt} = s_1 A_1e^{s_1 t} + s_2 A_2e^{s_2 t}

So A_1 and A_2 can be calculated from: 24 =A_1 + A_2 and 0 = s_1 A_1 + s_2 A_2 ?

Is this correct? It feels a little too simple. Also, is it alright to do i/C = dv/dt so that the C essentially goes away because if i = 0? Or should I do i = C dv/dt, insert the expression for dv/dt and multiply by C?

Thanks!

 

[rootX]

If initial condition for current is 0; it does not matter what you do with C as far it also not 0. 0/non-zeo or 0/C = 0