Give the derivation for LC oscillations & explain resonance with graph
When a capacitor is connected with an inductor, the charge on the capacitor and current in the circuit exhibit the phenomenon of electrical oscillations.

Let at t = 0, the capacitor is charged q m and connected to an inductor.

Charge in the capacitor starts decreasing giving rise to current in the circuit.

Let
q ? Charge
t ? Time
i ? Current
According to Kirchhoffï¿½s loop rule,
This equation is of the form of a simple harmonic oscillator equation. 
The charge oscillates with a natural frequency of and it varies sinusoidally with time as
Where,
? Maximum value of q
? ? Phase constant 
At
t = 0
q = ,
we have cos ? = 1 or ? = 0
? q = q m cos(?0t )
? i = i m sin?0 t
Where, i m = ?0 q m 
LC oscillations are similar to the mechanical oscillation of a block attached to a spring.

If a charged capacitor is connected across an inductor, charge will start to flow through the inductor, building up a magnetic field around it, and reducing the voltage on the capacitor. Eventually all the charge on the capacitor will be gone and the voltage across it will reach zero. However, the current will continue, because inductors resist changes in current, and energy to keep it flowing is extracted from the magnetic field, which will begin to decline. The current will begin to charge the capacitor with a voltage of opposite polarity to its original charge. When the magnetic field is completely dissipated the current will stop and the charge will again be stored in the capacitor, with the opposite polarity as before. Then the cycle will begin again, with the current flowing in the opposite direction through the inductor.