# The Application of Differential Equations in Business

In the following example we shall discuss the application of simple differential equation in business.

If $P$ is the principal amount and $r$ is the rate of interest, then the differential equation relating to the time $t$, $P$ and $r$ is

Example:

How long will it take for Rs.10000 to double if it is compounded continuously at 4 percent per annum?

Here $P = 10000$, $r = \frac{4}{{100}} = 0.04$

The differential equation governing the given problem is

Since the rate $r$ is fixed, putting the values of $r$ in equation (i), we have

Separating the variables in equation (ii), we have

Since the time during which $P$ is doubled is to be calculated, integrating the left hand side of equation (iii) from 10000 to 20000 and integrating its right hand side from 0 to $t$, we have

$\Rightarrow t = 17.328$ years, approximately.