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


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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!?

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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??

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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. :P 

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