Which one of these is correct!? (State technique for an electrical network)

[V8meathead]

Alright 2 of the same circuits.   One is from the signals and systems made ridiculously simple book that was posted on here.  The other one is from some professors notes I found online.  They get two different equations. Which one is correct and why!?  How do you figure this thing out, because I am confused as hell. 

The first picture is from the professor.
The second is from the book. 

They take a KVL equation around an RLC circuit and come up with two different equations.  Both assuming the same sign convention.  Maybe it's me that's screwed up.

Can someone help please!?

[V8meathead]

Click on them to view them better cause it's kind of hard to see the way it is.

Here is what im talking about.   2 Different equations.

[V8meathead]

If anyone needs the full version of the professors notes here they are...

http://www.engr.uky.edu/~ymzhang/EE422/EE422-7.doc

[indulis]

First, these to circuits are not exactly the same. They are however both 2nd order circuits. The professor is solving the system with state-space equations while the other solution is via differential equations. Since the circuits are different, why would you expect the solutions to be the same??

[V8meathead]

Thansk for your help!

I was mainly asking about how they did the current loops for the middle circuit. The main thing I was having trouble with was the defined capacitor polarity in the middle.  When I run my loops in a clockwise fasion like the professor I get different equations from that signals book.  But I think when you throw them into that matrix I'm pretty sure everything turns out to be alright.  The thing that was confusing me is that they ran their loops in 2 different directions for the middle loop, which is why you get 2 different equations. 

I was just curious as to why you define a capacitor polarity across the middle capacitor, but non of the other components.(I know you do it, I just can't remember why)  Im just having a brain fart!  I also realize it's a second order circuit.  Sadly, I've solved handfuls of these in class and can't remember the basics.  

[indulis]

From what I remember, and that was close to 30 years ago, it's just like writing mesh equations..... if you use the same conventions everywhere, the worst that can happen is the result, relative to your chosen polarity/direction will be negative.

[V8meathead]

Alright, awesome thanks