Exponential Growth

The Exponential Differential Equation

An equation is called a differential equation if it is an equation that contains derivatives.  In this section, we will consider differential equations of the type

dy/dt  =  ky

This differential equation can be interpreted as the equation that models the following statement,

"The rate of growth is proportional to the amount present."

Typically, solutions to differential equations are highly elusive.  However this differential equation has solution that are quite nice.

 Theorem (Exponential Growth)           y  =  Cekt is the solution to the differential equation           dy/dt  =  ky where C is some constant.

Proof

Let

y = Cekt

then

y'  =  kCekt  =  ky

A fundamental principle of nuclear chemistry states that the rate of decay of a radioactive element is proportional to the amount present.  In other words, the more you have, the more there is to lose.  Mathematically, if y is the amount present at time t, then

y'  =  ky

From the above statement, this differential equation has a solution

y  =  Cekt

for some constants C and k.  Notice that when we plug in 0,

y(0)  =  C

Hence, the constant C corresponds to the initial amount present.

Example

This year 500 pounds of Plutonium 239 (nuclear waste) was dumped in the Nevada desert.  The half life of Pu-239 is 24,360 years.  How long will it take until there is only 1 pound left?

Solution

Since

y'  =  ky

we can conclude that

y  =  Cekt

The initial amount present is 500 pounds, hence

C  =  500

The half life is 24,360 years so that y = 250 when t = 24360.  Plugging in gives

250  =  500ek24,360

1/2  =  ek24,360

ln(0.5)  =  24,360k

ln 0.5
k  =                  =  -2.95 x 10-5
24360

Hence

We set

and solve for t to get

t  =  218,407 years.

Exercise

Suppose that there is a fruit fly infestation in the central valley. Being an environmentalist, you propose a plan to spread 50,000 infertile fruit flies in the area to control the situation. Presently, you have in your laboratory 1,000 fruit flies. In 1 week they will reproduce to a population of 3,000 fruit flies. The farmers want to know when you will be ready to drop your infertile fruit flies. What should you tell them?