I have a RLC circuit where the capacitor is connected in parallel with a resistance and inductance in series. The battery is connected "in parallel" with the capacitor and the RL branches. At t=0 the battery is disconnected from the circuit. I need to find the voltage across each element using the Laplace transform.
Here’s what I did:
$$ E = RI + L\frac{dI}{dt} $$
Taking the transform I arrive at:
$$\frac{E}{s} = R ℒ(I) + L(s ℒ(I) - I_0)$$
With $I(t=0) = I_0$ $$ => ℒ(I) = \frac{E/L}{s(R/L + s)} + \frac{I_0/L}{R/L + s} $$
Which gives
$$I(t) = \frac{E}{R} + [I_0 - \frac{E}{R}]e^{-Rt/L}$$
The thing is this result looks kind of weird and I feel like I did something wrong. Could anybody let know what I should do about this?