Example 1:
A wet porous substance in the open air loses its moisture at the rate proportional to the moisture content. If a sheet hung in the wind loses half of its moisture during the first hour, when will it have lost?
(i) 95% moisture, weather condition remaining constant(ii) 90% moisture, weather condition remaining the same
Solution:
Let m0 be the moisture content initially and let m be the moisture content after t hours.
According to problem:
Substituting in (2), we have
\ Equation (2) becomes
(i) Again when the sheet losses 95% of the moisture, m = 
Equation (3) becomes
Example 2:
The decay rate of radium at any point t is proportional to its mass at that time. The mass is M0 at time t = 0. Form the differential equation and find the time when the mass will be halved.
Solution:
The differential equation for the decay is given by
where M is the mass of the radioactive substance after t hours.
= log M = - kt + log C
where log C is the constant of integration
(Substituting m = M0 t = to in (1))
M = M0 e-kt ….(2)
When the mass is halved,
Example 3:
Suppose the growth of a population is proportional to the population itself. If the population of a colony doubles in 50 days, in how, many days will the population become triple.
Solution:
Let the initial population be P0 and P be the population of the colony at any instant t.
Then according to the problem
log P = kt + log P0 ….(2)
when P0 is doubled,P = 2P0 where t = 50 days
From equation (2)log (2 P0) = 50 k + log P0
Example 4:
Find the equation of the curve that passes through the point (3,- 4) such that the slope of the tangent at the point (x, y) on it equals 
.
Solution:
According to the problem
Integrating, we have
y = x2c
This is the family of equations with parameter.
The family of curves is represented by y = x2c, since one of these curves passes through (3,- 4), we have
