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Differential Equations circuit homework problem

Attached is the problem from my text book. I'm confused about what I'm actually solving and how to use what I know. I know that Current for the inductor is given by the equation di/dt = v/L and the voltage of the capacitor is given by dv/dt = i/C

But how do I apply these to this system of equations. I tried just solving the system and seeing where that got me, but I come up with complex eigenvalues which leaves me with more confusion.

How do I use the picture and the known relationships to solve the system?
 

Homework problem: http://imgur.com/7lDYxzh

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1 Answer

Allison,
      There are 2 circuits, the one on the right is an RC circuit and the one in the left is an LCR circuit. If we start with the RC circuit and apply the loop theorem (conservation of energy) we get -iR-q/C=0 these are the voltage drops across the resistor and the capacitor  traversing the circuit clockwise. Now remembering that i=dq/dt this relation give a differential equation for the charge on the capacitor R*dq/dt +q/C=0.
     If you do the same for the RCL circuit we get iRR+Ldi/dt+q/C=0 where iRR is the voltage drop across the resistor, Ldi/dt across the inductor and q/C across the capacitor. Using i=dq/dt and di/dt= d2q/dt2 gives 
 
Ld2q/dt2+RRdq/dt + q/C=0 and the other equation is Rdq/dt+q/C=0 all in terms of the charge on the capacitor . Using these two fundamental equations and the definitions you should be able to derive the form listed in the reference document. Try it...
Jim