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?

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?