## Sketching Phase Curve Diagrams

Suppose we have the system of coupled differential equations
$\dot{x}=x+2y$

$\dot{y}=y$

Phase space diagrams illustrate the direction of the velocity field, in this example
$\begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}=\begin{pmatrix}x+2y\\y\end{pmatrix}$
.
At each point
$(x,y)$
draw the vector
$\begin{pmatrix}x+2y\\y\end{pmatrix}$
.

We can sketch curves - phase curves - to illustrate the velocity profile of the system by drawing smooth curves in the direction of the arrows at each point.