# 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

\[\frac{{dx}}{{dt}} \propto x\]

If $$k\,\left( {k > 0} \right)$$ is the proportionality constant, then

\[\frac{{dx}}{{dt}} = kx\]

Separating the variables, we have

\[\frac{{dx}}{x} = kdt\,\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]

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:

\[\begin{gathered} \int\limits_{400}^{800} {\frac{{dx}}{x} = k\int\limits_0^3 {dt} } \\ \Rightarrow \left| {\ln x} \right|_{400}^{800} = k\left| t \right|_0^3 \\ \Rightarrow \ln 800 – \ln 400 = k\left( {3 – 0} \right) \\ \Rightarrow 3k = \ln \frac{{800}}{{400}} = \ln 2 \\ \Rightarrow k = \frac{1}{3}\ln 2 \\ \end{gathered} \]

Putting the value of $$k$$ in (i), we have

\[\frac{{dx}}{x} = \left( {\frac{1}{3}\ln 2} \right)dt\,\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)\]

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:

\[\begin{gathered} \int_{400}^x {\frac{{dx}}{x} = \frac{1}{3}\ln 2\int_0^7 {dt} } \\ \Rightarrow \left| {\ln x} \right|_{400}^x = \frac{1}{3}\ln 2\left| t \right|_0^7 \\ \Rightarrow \ln x – \ln 400 = \frac{1}{3}\ln 2\left( {7 – 0} \right) \\ \Rightarrow \ln x = \ln 400 + \frac{7}{3}\ln 2 \\ \Rightarrow \ln x = \ln 400 + \ln {2^{\frac{7}{3}}} \\ \Rightarrow \ln x = \ln \left( {400} \right){2^{\frac{7}{3}}} \\ \Rightarrow x = \left( {400} \right)\left( {5.04} \right) = 2016 \\ \end{gathered} \]

Thus, there are 2016 bacteria after 7 hours.