Notes on Diffy Qs - Sage demos for section 1.6

Press the Activate buttons below to launch each Sage demonstration. After the demonstration launches you should be able to interact with it to change numbers with sliders. You may have to wait a little before the graph appears. Be patient.

Logistic equation

Consider the autonomous equation

\(x' = k x(M-x).\)

A plot of the slope field is shown, the two equilibria solutions are drawn in green. Three representative solutions are drawn in blue. Finally, a solution with the initial value \(x(0) = x_0\) is drawn in red. Sliders are presented for the values of \(k,\) \(M,\) and \(x_0.\)

Exponential growth/decay (with phase diagram)

Consider the general exponential growth and decay:

\(x' = k(x-A), \quad x(0)= x_0. \)

This demonstration will plot slope field, the phase diagram, and the solution of the given initial value problem in red along with the equilibrium solutions in green. The values of \(k,\) \(A,\) and \(x_0\) are changable with the sliders.

Logistic equation with harvesting (with phase diagram)

Let us add a constant to the logistic equation to allow harvesting:

\(x' = kx(M-x) - h, \quad x(0)= x_0.\)

The plot is the same idea as the demonstration for the exponential growth above.

The original code is mainly due to Ryan Burkhart.