State space representation of parllel RLC circuit
There are several possible state variable presentations for the same circuit, but I guess you want state variables which are measurable quantities of the circuit. In that case you should use inductor currents and capacitor voltages as the state variables.
Capacitors have stale law dUc/dt = I/C where Uc is the cumulated voltage over the capacitor and I is the charging current. Inductors have dI/dt = U/L where I is the cumulated current through the inductor and U is the voltage over the inductor.
The series RLC circuit:
You can write the wanted equations with these laws. Express the voltage over the inductor with your V(t), I, R and Uc. Divide it by L and there's your expression for dI/dt, the right side of the first state equation.
The second state equation is simply (as already written) dUc/dt = I/C.
Circuit analysis programs which do transient analysis generate the equations automatically and solve them numerically. You must enter the initial values of the state variables i.e. inductor currents and capacitor voltages or let them be zero.