Mathematica: Stream plots for differential equations

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

Stream plots for a single equation

Consider the example

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

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

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

Using the differential equation, we see that

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

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:

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

For a system of the form

$\dfrac{dx}{dt} = f(x,y) \\ \\ \dfrac{dy}{dt} = g(x,y)$

we can make a stream plot on the domain $a\leq x \leq b$ , $c\leq y\leq d$ using


StreamPlot[{f(x,y),g(x,y)},{x,a,b},{y,c,d}]

For example, the stream plot for the system

$\dfrac{dx}{dt} = x(1-x) + \sin{y} \\ \\ \dfrac{dy}{dt} = y^2 + \frac{x}{e^y}$

is generated using the code


StreamPlot[{x(1-x)+Sin[y], y^2+x/Exp[y]},{x,-2,2},{y,-3,3}]

The result is

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