That's the problem and it's a first order transient circuit problem.
What I'm trying to find is $v_0(t=0)$ and $v_c(t=0)$ values.
The result for $v_0(t=0)$ is 5V.
What I know is, there's a steady-state for capacitor before t=0.
So the capacitor behaves as an open circuit for t<0.
In this case, are the top left 1ohm and 2ohm resistors are in series?
I mean there's no current through the capacitor, so the current from 12V voltage source must flow through 1ohm resistor and then 2ohm resistor, right?