Solve Freq of Coupled LC Circuit with Inductor

[aseylys]

Homework Statement

I have to find the normal frequencies of a coupled LC circuit. However, this LC circuit is coupled by an inductor, not a capacitor.
__|C|________|C|__
|...|...|
^I(1)...|...^I(2)
|...|...|
{L}...{L'}...{L}
|...|...|
|...|...|
--------------------

I'm sorry, I didn't have a picture but that's basically the circuit.

Homework Equations

I₁+I₂=0
I=dq/dt

These equations are for an LC circuit coupled by a capacitor:
L(d²q₁/dt²)+(1/C)q₁+(1/C')(q₁+q₂)=0

L(d²q₂/dt²)+(1/C)q₂+(1/C')(q₁+q₂)=0

The Attempt at a Solution

The only attempt I could figure out that it would be similar to that of a circuit coupled by a capacitor. I don't know if I'm on the right track or not and if I am I'm not sure how to modify the two equations.

 

[tman12321]

I1 + I2 = 0 is no longer correct. Current flows through the inductor.

 

[rude man]

I see an L' in your image but not in your equations. Conversely I see a C' in your equations but not in your image.