First of all, since you are not combining the two storage elements L1 and L2, you have a second order circuit analysis instead of a first order circuit like you claim.Where is the error in this analysis?
http://www.ece.uwaterloo.ca/~ece204/TheBook/14IVPs/higher/complete.html, under "RLL Circuit"From your circuits course, you can determine that the differential equation describing the current flowing across the second inductor is given by:
y(2)(t) + (10/3 + 10/2)y(1)(t) = -10/2 sin(t)
Rewrite this as a system of 1st-order differential equations.
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