I'm having some trouble figuring out how to draw (by hand) the vector field given a set of differential equations. Consider the following:
$\frac{dx}{dt} = x+y$
$\frac{dy}{dt} = -x + y$
Normally, when I am given just one differential equation, like $\frac{dy}{dt} = y$, I can easily compute the values by hand and can plot this out – think of this as picking coordinates of $(t,y)$. Would this approach be the same for this given system of differential equations? I'm more interested in the process, but a graph of how this should look like would be greatly appreciated as well.
Best Answer
If $x(t)=a$ and $y(t)=b$ for some time $t$, then the "arrow" emanating from the point $(a,b)$ points in the direction $(a+b,-a+b)$. This is the vector field to be drawn. For example, using Mathematica,
gives