Working on a problem involving differential equations. My knowledge of DE's is pretty limited, as I am taking that class this semester too, and they have no coincided yet.

Anyways, looking for some feedback about this problem. Where I am going wrong, maybe?

I have attached thumbnails of the given problem, and my work (the closest I've gotten yet).

I had a problem very similar involving RC and I solved that one fairly easily, but this one is throwing me.

My assumptions and method are as follow:

In the problem statement, The switch is in position 1 for a long time, then at t=0 the switch is thrown to position 2.

In that way, I assume there is nothing to account for in the center branch from the middle top node--so therefore I have accounted for nothing in that branch in my node equation. I know that after a long time L becomes a short, or loses all resistance.

Maybe nodal isn't the way go this time? Maybe I should be doing a mesh around the whole outside loop?

Thanks in advance.

Given:

My work: