Before solving this question i have some doubts i need to clarify.

I been learning RC/RL Circuit and RCL Circuit.

So RC Circuit are classified as First Order and RCL Circuit are Second Order?

What it means Source Free RC/RL Circuit and Step Response RC/RL Circuit?

Does it means that at t<0 the Capacitor/Inductor is up and when t>0 it starts to decay without the source?

If my above analogy is correct, in order to know whether it is Source Free or Step Response Circuit, we just have to see if there is source attached to it when t>0?

Now to what i done so far for the question:

i got the steps from referencing my textbook.

At time = 0-

i(0+) = i(0-) = 8/2 = 4A ---- why i(0+) = i(0-)

v(0+) = v(0-) = 2i(0-) = 8V --- why v(0+) = v(0-)

α = R / 2L = 2/2 = 1

ω(o) = 1/ sqrt(LC) = 1 / sqrt [(1)(1/5)] = 2.236 -- what is this α and ω representing? Is the formula being use for all RCL Circuit analysis?

S(1,2) = -1 ± √[(1^2) - (2.236^2)] = -1 ± j2

Since α < ω(o) it is underdamped. ---- what is underdamped,critically damped and overdamped.

v(t) = (A1 cos ωt + A2 sin ωt) e ^(-αt) + Vs

dv/dt = -A1 + 2A2

dv(0)/dt = 4 / C = 20 ---- why we need get the dv/dt and dv(0)/dt to compare?

Now im stuck on how to find the V(0) .

when t<0 the voltage source supplied to the circuit is 8v and t>0 is 12v.

but i do not know how to find the voltage across the capacitor which is series with e voltage source, inductor and cap.

It is alot of question. But i hope someone could explain these to me as im very confused with this new stuff being taught to me.