Could someone please explain how you draw the direction field and plot a few trajectories of a system. An example would be
x'= {3 -2 } x
{2 -2 }
In brackets is a 2 by 2 matrix if its not obvious. Any help appreciated greatly.
Could someone please explain how you draw the direction field and plot a few trajectories of a system. An example would be
x'= {3 -2 } x
{2 -2 }
In brackets is a 2 by 2 matrix if its not obvious. Any help appreciated greatly.
$\displaystyle X'= \begin{bmatrix}3 & -2 \\ 2 & -2\end{bmatrix}X$
or, if $\displaystyle X'= \begin{bmatrix}x \\ y\end{bmatrix}$,
$\displaystyle \begin{bmatrix}x' \\ y'\end{bmatrix}= \begin{bmatrix}3 & -2 \\ 2 & -2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}$
which is the same as the system of equations, x'= 3x- 2y, y'= 2x- 2y.
For a number of different points (x,y), calculate x' and y' and, at that point, draw a short vector with slope y'/x'. That's the "direction field". A "trajectory" is a continuous curve that is parallel to the direction field vectors at each point.