A Phase Portrait is one type of qualitative tool for studying first order systems of ODEs. For the first order system,
x' = f(x,y)
y' = g(x,y),
Here is an alternate way of thinking about it. We think of the independent variable t as the time variable and the vector (x(t),y(t)) as representing the position of the solution curve in the xy-plane at time t. As t varies, the point (x(t),y(t)) ``moves'' along the solution trajectory in the xy-plane. The velocity at which the point moves is given by the vector of derivatives (x'(t),y'(t)) (we sometimes use the notation (dx/dt,dy/dt) in class). First, we know that the given first order system holds. If we choose a point (x,y) in the xy-plane, then the velocity vector for the solution curve passing through that point in the plane is given by (x',y')=(f(x,y),g(x,y)). That is, we can pick a point in the xy-plane, evaluate the functions on the right hand side of the system at that point, and these values determine the velocity vector of the solution curve at that point. When we plot the phase portrait, we simply compute and diagram a variety of velocity vectors in the plane and sketch solution curves by following along the velocity vectors.
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