# RLC series circuit with initial current

I'm trying to solve, for the voltage across the capacitor in the following circuit:

simulate this circuit – Schematic created using CircuitLab

The initial current in this circuit is equal to $$\\text{I}_0\$$. How can I find the voltage across the capacitor?

EDIT: The initial applied current is equal to $$\\text{I}_0\$$, the initial voltage across the capacitor is equal to $$\\text{V}_\text{C}\left(0\right)\$$ and the initial current trough the inductor is equal to $$\\text{I}_\text{L}\left(0\right)\$$

My Work:

I thought that I can write:

$$\text{V}_{\text{R}_1}\left(t\right)+\text{V}_{\text{R}_2}\left(t\right)+\text{V}_\text{C}\left(t\right)+\text{V}_\text{L}\left(t\right)=0\tag1$$

Writing that in terms of the current through the components I get:

$$\text{I}_\text{in}'\left(t\right)\cdot\text{R}_1+\text{I}_\text{in}'\left(t\right)\cdot\text{R}_2+\text{I}_\text{in}\left(t\right)\cdot\frac{1}{\text{C}}+\text{I}_\text{in}''\left(t\right)\cdot\text{L}=0\tag2$$

And then using the initial conditions $$\\text{I}_\text{in}\left(0\right)=\text{I}_0\$$ and $$\\text{I}_\text{in}'\left(0\right)=0\$$ I can solve for the voltage across the capacitor. Is that right?

• Is current $I_0$ supplied from a current source and was the current through the inductor prior to t=0 zero amps AND, was the voltage across the capacitor 0 volts at t=0? In other words you need to clearly state your initial condictions. At the moment there is ambiguity. – Andy aka Dec 20 '18 at 18:46
• @Andyaka Yes only the intial current was supplied. – Klopjas Dec 20 '18 at 18:47
• That does not answer what I asked. – Andy aka Dec 20 '18 at 18:49
• @Klopjas You must have the initial conditions for the state of the capacitor and the inductor. Given only the value of $I_0$ I believe there is more than one possible pairing of initial conditions for the capacitor and inductor. – jonk Dec 20 '18 at 18:51
• @Klopjas then all that remains for you to clarify is the source of that external current because that source type dictates what happens after t=0. – Andy aka Dec 20 '18 at 18:53