Phase Plane Methods

this automatically executes and loads the package needed for the direction field.

dx/dt=f[x,y]
dy/dt=g[x,y]

Clear[f,g]
f[x_,y_]:=y^2 - x^2
g[x_,y_]:=x-1

nullclines

Solve[f[x,y]==0]

Solve[g[x,y]==0]

equilibria

Solve[{f[x,y]==0,g[x,y]==0}]

graphing window containing interesting places

Clear[xmin,xmax,ymin,ymax]
xmin=-2.0;
xmax=2.0;
ymin=-2.0;
ymax=2.0;

plots of nullclines and equilibria

Clear[np1,np2,np3,np]
np1 = ParametricPlot[{-t,t},{t, ymin, ymax},PlotStyle->{Blue,Dashed,Thick}];
np2 = ParametricPlot[{t,t},{t, xmin, xmax},PlotStyle->{Blue,Dashed,Thick}];
np3 =  ParametricPlot[{1,t},{t, ymin, ymax},PlotStyle->{Red,Dashed,Thick}];
np = Show[{np1,np2,np3,np4},PlotRange->{{xmin,xmax},{ymin,ymax}}];
Clear[pts]
pts = Graphics[{PointSize[Large],Point[{{1,-1},{1,1}}]}];
Show[np,pts]

the direction field

Needs["VectorFieldPlots`"];
Clear[vPlot]
vPlot = VectorFieldPlot[{f[x,y],g[x,y]},{x,xmin,xmax},{y,ymin,ymax}]

all together

Show[np,pts,vPlot]