Examples of Simple 2X2 Linear Systems - Direction Fields

Here are a variety of some simple linear, autonomous, homogeneous 2X2 systems of ordinary differential equations. Various plotting options are demonstrated.

Students should be able to identify the system, identify nullclines and equilibria, and sketch some trajectories. 

This notebook was created by T. Shelton, Southwestern University to supplement a handout for  Differential Equations II, updated for Mathematica 7.0.

reference  Differential Equations: Theory, Technique, and Practice, Simmons & Krantz, McGraw-Hill Co, Inc. 2007

Clear[xmin,xmax,ymin,ymax]
xmin=-3.0;
xmax=3.0;
ymin=-5.0;
ymax=5.0;

VectorPlot[{1,1},{x,xmin,xmax},{y,ymin,ymax},Axes->True,VectorScale->{Tiny}]

VectorPlot[{0,y},{x,xmin,xmax},{y,ymin,ymax},Axes->True,VectorScale->{Small},VectorColorFunction->Hue]

VectorPlot[{x,0},{x,xmin,xmax},{y,ymin,ymax},Axes->True,VectorScale->{Small},VectorColorFunction->Hue]

VectorPlot[{2x,-y},{x,xmin,xmax},{y,ymin,ymax},Axes->True]

VectorPlot[{y,-x},{x,-3,3},{y,-3,3},VectorScale->{Automatic,Automatic,None},Axes->True]