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# Notes on Diffy Qs - SAGE demonstrations for section 1.6

Press the *Activate* buttons below to launch the SAGE demonstration.
After the sage cell is evaluated 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

Let us start with the autonomous equation

$x' = -k x(x-A)$.

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

## Exponential growth/decay

Let us now look at the exponential growth and decay:

$x' = k(x-A)$, $x(0)= I$.

This cell will plot the above exponential growth and decay. The values of $k$,
$A$, and $I$ are changable with the sliders. In this plot, we will also
draw the phase diagram to the right of the plot.

## General quadratic

Let us add a constant to the logistic equation

$x' = -kx(x-A) + B$, $x(0)= I$.

Again we will plot the phase diagram on the right of the plot.

The code is mainly due to Ryan Burkhart, it was then slightly modified by Jiří Lebl.