Differential equations are frequently used in solving the mathematical and physical problems. In the following example we shall discuss application of simple differential equation in biological sciences.
In a culture bacteria increases at the rate proportional to the number of bacteria present. If bacteria are 400 initially and are doubled in 3 hours, find the number bacteria present 7 hours later.
Let be the number of bacteria, 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 bacteria are 400 initially and doubled in 3 hours, so we integrate 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 this 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, 2016 is the required number of bacteria after 7 hours.