And the suggestion was to write down the ODEs instead of using the phasor methods.

Note that when you are using phasors you are really working in the frequency domain, meaning the formulas aren't very useful if you are interested if is happening as a function of time (which, since you are asking about lag, you are).

Now, if you do take the time to write down the ODEs you will see that inductors are essentially

*integrating* elements meaning what happens in them depends on their "history", this is why it takes time for them to "react" and is causing the phase shift. Now, if you want to understand the reason for that you need the full EM theory (Maxwell's equations) for real inductors, this is presumably beyond the scope of a course in circuit theory.

Asking "why something is REALLY happening" is -as is usually the case in physics- a rabbit hole.

Do you have access to some software that can be used to solve ODEs as a function of time? If you do I would suggest you spend some time solving the ODEs for your circuit different initial conditions etc looking at the instantaneous values for voltage or current, this is -in my experience- the best way to get an intuitive grasp of what is happening.

Or, alternatively, you can just download a free circuit simulator such as LTSpice (see

http://www.linear.com/designtools/software/ ,very useful piece of software). The advantage here is that you can "probe" different parts of the circuit and see voltage, current etc.

Since it is such a simple circuit you are dealing with you can of course get the same information from the formulas but using a numerical simulator leaves you more time to think about the physics (since you don't have to worry about the math)