Rami Arieli: "The Laser Adventure" Chapter 2.8 page 1
2.8 Rate Equations for spontaneous emission

(Advanced level only, need to know first order differential equations).

For simplicity we shall assume:

The rate at which the excited atom population N2(t) decays from the higher energy level (E2) to the lower energy level (E1), by stimulated emission is given by the decay coefficient g21 multiplied by the instantaneous population number of this level N2(t) as seen by the first order differential equation:
d N2(t)/dt = - g21 N2(t) = - N2(t)/ t2

This equation defines the lifetime t2 of energy level E2:

t2 = 1/ g21

(For a reminder about the solution for a differential equation of this kind, Click here)

The solution to the rate (differential) equation is:

N2(t) = N2(0) exp(-g21t) = N2(0) exp(-t/ t2)

Question 2.5:
check by substitution that the solution is valid.