If an inductor is connected to A.C. source (thinking no resistance is there), and if we travel across circuit there is voltage drop, and it is said
$$V=L\frac{\text dI}{\text dt}$$
How we write this equation? And if the applied voltage equals back emf, how does current flow?
The back EMF is created by the current flow, via the associated magnetic field. It rises until it equals the applied voltage. The difference is then zero and the current can no longer increase. But it is still flowing.
The relationship between voltage and current for a given coil is called its impedance $Z$. It is strongly frequency-dependant, but that is another story.
