Inductor Coupling Question
Ideal Transformer 
By Faraday's law of induction:

Vs = − Ns dΦ / dt 
Vp = − Np dΦ / dt

Where V is the instantaneous voltage, N is the number of turns in a winding, dΦ/dt is the derivative of the magnetic flux Φ through one turn of the winding over time (t), and subscripts p and s indicate primary and secondary.

Combining the ratio of equations 1 and 2:
Turns ratio = Vp/Vs = −Np/−Ns 

Now to your question: 
You can not simply determine i1 with the inductor equation V1=L*(di1/dt) on the primary side because i1 is also dependent on the magnetic flux through both windings and i2 on the secondary side. Also, L2 is missing in your equations.