In the following circuit the switch is closed at `t=0`.Intially there is no current in inductor.Find out current the inductor coil as a function of time.

Explanation

At any time `t`
`-epsilon+i_(1)R-(i-i_(1))R=0`
`-epsilon+2i_(1)R-iR=0`
`i_(1)=(iR+epsilon)/(2R)`
Now `-epsilon+i_(1)R+iR+L.(di)/(dt)=0`
`-epsilon+((iR+epsilon)/2)+iR+L.(di)/(dt)=0 rArr -epsilon/2+(3iR)/2=-L.(di)/(dt)`
`((-epsilon +3R)/2)dt=-L.di rArr -(-dt)/(2L)=(di)/(-epsilon +3R)`
`-underset(0)overset(t)int(dt)/(2L)=underset(0)overset(t)int(di)/(-epsilon+3iR) rArr -t/(2L)=1/(3R)"In"((-epsilon+3iR)/-epsilon)`
`-"In"((-epsilon+3iR)/-epsilon)=(3Rt)/(2L) rArr i=+epsilon/(3R)(1-e^((3Rt)/(2L)))`