Sho Kano said:
The current through L2 just after the switch is closed is also zero.
If the initial current (through L1) is i, it will split up into two currents at the junction, i1 into R2, and i2 into L2.
i = i1 + i2
i1, the current through R2 is = i - i2
Sho Kano said:
Ah I see he's just expressing it in terms of the other currents right?
Right.
The "+" or "-" sign that goes in front of a particular current depends on how you set up and define your currents at the beginning.
Back in your Post #4, it seems that you used Kirchhoff's Voltage Law (KVL, as opposed to KCL) to define your currents in the form of current loops. You get to define these loops yourself. (There are multiple ways to define the loops in KVL. It's your job to choose one and be consistent from then on.)
You may have made a mistake in there somewhere when setting up your equations, btw (in post #4).
But using KVL (instead of KCL) to set up and define your current directions is perfectly okay. Once you define your currents, the important part after that is to maintain consistency.
Whether you use KCL or KVL to set up your system of equations, the final results should produce identical answers in the end, when evaluating the current through and voltage across any given component.
You are the one in charge of setting these things up. And you must maintain self consistency.
So yes, i = i_1 + i_2 is correct, or at least can be correct,
depending on how you define i, i_1 and i_2.
In KVL it depends on which loops pass through a given component, and the direction of each of the loops (clockwise or counterclockwise).
In KCL it depends on which currents are directed into a node and which currents are directed out of a node.