Are you just looking for a description of the waveforms in an LC circuit? What you have to realize is that analyzing circuits with inductors and capacitors in them is no different than analyzing a circuit with resistors. It's done exactly the same way- the equations for the voltage and current for inductors and capacitors is just more complicated than resistors is all, and make it more involved to solve the equation.
Like Miles says, it is a loop analysis of an LC circuit.
In the same way:
resistors are
V=IR
inductors are
V = L*di/dt
capacitor are
I = C*dv/dt -> dV = (I/C)dt -> V = ∫(I/C)dt
If you just have an L and a C connected in a loop, Kirchoff's voltage law around the loop works out to be:
0 = Vinductor - Vcapacitor
0 = L*dI/dt - ∫(I/C)dt
This is a differential equation, but is not in the form we are used to solving (for the form required by most methods to solve the equation) because it has an integral in it. The form we are used to solving has all derivatives in it. So to change this equation into that form we take the derivative of both sides.
0 = L*d^2I/dt^2 - I/C
Now it has all derivatives and is almost in the form we want. We can solve it in this form now, but usually we like to remove the coefficient from the term with the highest derivative (some methods to solve require it be like this, others do not). We divide everything by L to remove the L from the term of the highest order (the term with the second derivative):
0 = d^2I/dt^2 - I/LC
This differential equation is now in a form that matches the form required by most methods used to solve it. This is a differential equation where the variables are I and and it's derivative, dI/dt. You need to solve it using partial differential equation methods (Laplace transforms is one method to solve this equation). You simulate the disconnection of a power source by setting the initial conditions in your solution to be that where the capacitor is charged up to a certain voltage or where the inductor has an initial current flowing through it.
THe solution you get for I and dI/dt will be an equation that describes the time behaviour current in the circuit. Plugging it back into the Capacitor and Inductor voltage formulas will give you their voltage behaviour over time. Because it is an oscillator your solution will be some kind of periodic waveform (for both current and the voltages of the capacitor and inductor).
An example of this math that is carried out to completion is under "Time Domain Solution" is one example of how to approach the problem. This particular method does not use Laplace transforms to solve it. They leave out a lot of details on what goes on as they solve the differential equation, but a lot of people prefer Laplace transforms instead. If you do not understand what they are doing you need to learn how to solve differential equations- that is the key.
https://en.wikipedia.org/wiki/LC_circuit