Why is lenz law not applied in this crcuit analysis
Lenz's Law says that the direction of an induced current is always such as to oppose the change in the magnetic field that produces it. 

In your circuit the current producing the magnetic field is entering the coil at the '+' end, so the induced current must be flowing in the opposite direction, out of the '+' end. For current to be pushed out of that end the induced voltage must be positive, not negative.