# What is the differential equation that models exponential growth and decay?

##### 1 Answer

The simplest type of differential equation modeling exponential growth/decay looks something like:

#dy/dx = k*y#

This differential equation is describing a function whose rate of change at any point

#y = C * e^(kx)#

where

Just to demonstrate how this works, let's say that we have a droplet of water being absorbed into a piece of cloth. At any given moment, the droplet of water is shrinking by 10% of its current size. We want to find a function,

This situation translates into the following differential equation:

#dy/dt = - 0.1 * y#

First step in solving is to separate the variables:

#-1/(0.1y) dy = dt#

Now, we will simply integrate:

#int -1/(0.1y) dy = int 1 dt#

The right side is fairly easy. Remember the constant of integration:

#int -1/(0.1y) dy = t + C#

Note that we can pull

#-1/0.1 int 1/y dy = t + C#

And now this is easily solved:

#-1/0.1 ln y = t + C#

Now, we will multiply both sides by

#ln y = -0.1t + C#

Exponentiate both sides:

#y = e^(-0.1t + C)#

This can be rewritten as:

#y = e^C * e^(-0.1t)#

Again, since

#y = C * e^(-0.1t)#

And there is our equation for the size of the droplet at time

#100 = C * e^(-0.1*0)#

#100 = C#

#y = 100 * e^(-0.1t)#

There we go. If you graph this function on your calculator, you can verify that it does indeed have the property that at any point