## Mathematica: Stream plots for differential equations Paul T. Allen

It is very easy to use Mathematica to make stream plots for differential equations.

## Stream plots for a single equation

Consider the example

\$latex

frac{dy}{dt} = (y-y^3)cos{(t)}.

\$

Points on a solution curve to this equation will take the form \$latex (t,y(t))\$. Thus the slope will look like

\$latex (frac{d}{dt}[t],frac{d}{dt}[y(t)]) = (1,frac{dy}{dt})\$.

Using the differential equation, we see that

\$latex

left((1,frac{dy}{dt}right) = left(1, (y-y^3)cos{(t)}right).

\$

Thus this is what we want to plot. The Mathematica code for this is the following:

StreamPlot[{1,(y-y^3) Cos[t]}, {t,0,5},{y,-2,2}]

The output is this: Notice the following:

• The command `{1,(y-y^3) Cos[t]}` gives the slope to be plotted at each point.
• The command `{t,0,5}` gives the range of \$latex t\$ values.
• The command `{y,-2,2}` gives the range of \$latex y\$ values.

Notice also that it is easy to see the equilibrium solutions on the picture – how cool is that!

## Stream plots for systems

For a system…

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