Symmetry

    The solution curve to the logistic differential equation

            .  

is given by

            .  

and it is symmetric about the point  .

Proof.

What would it take to convince us of this fact?

We can easily determine that lies on the curve .

We can substitute into the formula  .

And we can look at the graph in the example to see if it "looks right."

This might be easier to see if we plot the curve over an interval symmetric about  .

The following animation might make this fact easier to see.

The general proof for symmetry requires a bit of effort and thought. We want to solve the D. E.

    .

With the initial condition

    .

And we know that it is to pass through the point of symmetry

    .

So why not ask Mathematica to do it !

And now we can determine if  f[t]  is symmetric about the point .

The point of symmetry can be shifted to the origin as follows:

We must verify that  g[t] = -g[-t] .

Therefore, we have shown that the logistic curve is symmetric about the point    .

It is easier to see this fact graphically.

We must verify that  g[t] = -g[-t] .

(c) John H. Mathews 2004