# The Application of Differential Equations in Biology

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.

Example:
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 $x$ be the number of bacteria, and the rate is $\frac{{dx}}{{dt}}$. Since the number of bacteria is proportional to the rate, so

If $k\,\left( {k > 0} \right)$ 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 $k$ as follows:

Putting the value of $k$ 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 $x$ and its right side from 0 to 7 as follows:

Thus, there are 2016 bacteria after 7 hours.