As you can probably see there are many methods one can apply to solving the problem you posted.@MrAI the switch is open for a long time and it closes on t=0.
@t_n_k Thanks a lot, i understand now how things are, what i dont understand though is why the equivalent without the source is wrong, is that a wrong practice ? Do you think i am unaware of a crucial theory path ? If yes can you give me a link to study about it ?
My impression is that you may have been taught a general method which is based on a standard form solution for 2nd order systems. Your teacher may be bypassing some of the details of a methodical solution from first principles. There's not much point in attempting to teach you something new - rather it would be better if you reviewed what you have been taught to ensure you have fully grasped the concepts.
Also keep in mind that an equivalent circuit must exactly reflect the true circuit conditions with all driving soures accounted for. Removing a driving source may allow one to define the natural circuit response parameters but one can't substitute driven responses (such as the inductor current response in your example) back into the undriven equivalent circuit and expect to obtain a sensible answer.
Also more by way of a general comment, an equivalent circuit can in some circumstances "hide" one or more circuit variables by virtue of the reductions applied.