Differential equations are frequently used in solving mathematics and physics problems. In the following example we shall discuss the application of a simple differential equation in biology.
In a culture, bacteria increases at the rate proportional to the number of bacteria present. If there are 400 bacteria initially and are doubled in 3 hours, find the number of bacteria present 7 hours later.
Let be the number of bacteria, and the rate is . Since the number of bacteria is proportional to the rate, so
If is the proportionality constant, then
Separating the variables, we have
Since there are 400 bacteria initially and they are doubled in 3 hours, we integrate the left side of equation (i) from 400 to 800 and integrate its right side from 0 to 3 to find the value of as follows:
Putting the value of in (i), we have
Next, to find the number of bacteria present 7 hours later, we integrate the left side of (ii) from 400 to and its right side from 0 to 7 as follows:
Thus, there are 2016 bacteria after 7 hours.