Parallel transformers - checking my models

[fahraynk]

So on the left is a circuit which shows a voltage source and 2 coupled inductors in a parallel magnetic circuit shown on the right (transformer with a parallel core).
Can someone tell me if I am modelling it correctly with the following equations?
Rc1,Rc2,Rc3 are the reluctance of each branch of the transformer core
M = mutual inductance between the coils.
φ1 and φ2 are the flux on left loop and right loop of the magnetic circuit.
φ1₀ and φ2₀ are the total flux from L1 and L2 alone
$$M=k\sqrt{L1*L2}\\k=\frac{\phi_{12}}{\phi{1_0}}=\frac{\phi_{21}}{\phi{2_0}}$$

Here are the equations I came up with :

$$
MI_2'+L_1I_1'+R_1I_1+V_0cos(wt)=0\\
MI_1'+L_2I_2'+R_2I_2=0\\
\phi_1R_{c1}+R_{c2}(\phi_1-\phi_2)=N1I1\\
\phi_2R_{c3}+R_{c2}(\phi2-\phi1)+N1I1-N2I2=0\\$$
so the total flux through path 2 (φ₂)- the total flux produced by the 2nd inductor (φ2₀)= φ₁₂

$$L_2I_2=N2\phi2_0\\k=\frac{\phi_{12}}{\phi1_0} \hspace{10 mm}\phi_{12}=\phi_2-L_2\frac{I_2}{N2}\\
k=\frac{(N2\phi_2-L_2I_2)N1}{N2L_1I_1}$$

Is this enough equations to solve... or am I missing an equation? If so... what equation am I missing? I think there needs to be something relating I2 to I1 without the derivatives... ?

 

[fahraynk]

Tried to pay a tutor to answer but they said I don't have to pay because their answer might be wrong -_-

 

[valentin bogatu]

This man isn't serious, he got his answer, never tried to pay the tutor...

 

[fahraynk]

This man isn't serious, he got his answer, never tried to pay the tutor...

I tried to pay for an answer, meaning If I got an answer for the current I1, I2. But after like 48 hours I went through like 5 people and did not get the answer? I pretty much got my own work shown back to me with no current.