Separate the variables in the logistic differential equation
Then integrate both sides of
the resulting equation. (This is easy for the "t" side -- you may
want to use your helper application for the "P" side.)
After calculating both integrals, set the results equal. If
your helper application does not know about constants of integration,
provide one in an appropriate place in your equation. Then solve the
equation for P as a function of t.
Now use your helper application's differential equation
solver to solve the logistic equation directly. If the resulting equation
is not already solved for P as a function of t, use an
additional "solve" step to complete the symbolic calculation.
The results from steps 2 and 3 are
-- or should be -- formulas for the same family of functions. If the
formulas do not look alike, reconcile any differences that you see. Show
that one formula can be put in the form of the other. If the formulas have
"arbitrary constants" in different places, show how the arbitrary constant
in one formula is related to the arbitrary constant in the other
In your integration in step 2, you
may have encountered ln(P) and ln(K - P), both of which make
sense if P is between 0 and K. But the second one does
not make sense if P > K, so your formula may not be correct in this
case. Change your equation in step 2 to one that would be correct if P >
K, and solve again for P. Reconcile the result with the form
generated by the differential equation solver in step 3, if possible. If
you think the form generated by the DE solver does not work for P >
K, explain why.
starting population P(0) is a specific number P0
(which may be either smaller or larger than K). Choose whichever
solution form you prefer, and determine the value of the "arbitrary"
constant (in terms of K and P0) that produces a
solution P(t) such that P(0) = P0. Simplify as
much as possible.
Explain why your
solution function P(t) in step 6 approaches K as t
For specific values
of r, K, and P0, plot the direction field
and your solution function to verify visually that your formula is
correct. Repeat with several combinations of the parameters, including
values of P0 both smaller and larger than K.