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