Background

    The exponential model    is used to study uninhibited population growth and solution is the exponential function  .  When the term   is added we obtain the logistic differential equation which is used to model inhibited population growth or bounded population growth.  The logistic differential equation is

            .  

One form of the solution is  

            .  

The terms have been carefully determined so that the initial condition is

            .

The limiting value  L  of  y(t)  is given by

            .

The graph is the so called "S-shaped" curve. The choice of parameters    creates the curve shown below.

Proof.

The traditional pencil and paper for finding the solution to the differential equation    uses the technique of separation of variables.

We can use Mathematica to simulate this derivation.

Move  dt  to the right side.

Isolate  dt  on the right side.

Integrate both sides.

Expand the left side.

Take the anti-logarithm of both sides.

Simplify the left side.

Solve for y[t].

Of course we could have used Mathematica's "Solve" procedure at step 5 and saved a lot of "hand computation."

Aside.

If we wanted to see all the results in those hand computations we could list them.  This is just for fun !

Store the solution in  f[t].

We might like to fuss around with the constant a bit, the following replacement and expansion makes things nicer.

We can have Mathematica verify the solution by "plugging" it into the D. E.

Notice that if  ,  then  ,  and  .   
In applications this is often the situation.  

If you prefer to have the initial condition , then it is easy to determine so that this occurs.

Which we can manipulate by hand and write as  

This form of the solution has the desired initial condition.

We can also find the limiting value as by using the "Assumption" that  0 < a ,  and evaluating  .  

The graph at the top was made with the choice of parameters  .

Aside.

Of course we could have used Mathematica's "DSolve" and obtained the general solution.

But the solution is not in the form we desired.

    .  
    
It would still take some clever manipulations to change the form of the solution.

The initial value problem could be solved with Mathematica too.
At first glance one might try the command:

Woops!  What went wrong?

Mathematica requires that we use precise mathematics and use fractions instead of decimals.

Don't worry about some of the error messages in Mathematica, after all we should agree that it is "hard" to solve complicated equations.

We can substitute fractions for those decimals to see that the solutions are the same.

So we see that the two solutions are the same.

(c) John H. Mathews 2004