Example with

> ode1 := diff( x(t), t ) = -2*x(t) + y(t);

> ode2 := diff( y(t), t ) = x(t) - 2*y(t);

> MODEL := { ode1, ode2 }:

> VARS := { x(t), y(t) }:

> DOMAIN := t=0..1:

> RANGE := x=-2..2, y=-2..2:

> dirPLOT := DEplot( MODEL, VARS, DOMAIN, RANGE, arrows=MEDIUM ):

> dirPLOT;

>

Can you SEE both of the straight-line solutions in this direction field? Let's see if this is consistent with the eigenvalues and eigenvectors analysis:

> A := matrix( [ [ -2, 1 ], [ 1, -2 ] ] );

>

> charpoly( A, lambda );

> eigenvals( A );

> eigenvects( A );

>

Thus, the straight-line solutions are:

> Y1 := exp(-3*t) * matrix( 2, 1, [ -1, 1 ] );

> Y2 := exp(-t) * matrix( 2, 1, [1,1] );

>

> ICline := [ [ x(0)=2, y(0)=2 ], [ x(0)=2, y(0) = -2 ], [ x(0)=-2, y(0)=2 ], [ x(0)=-2, y(0)=-2 ] ]:

> linePLOT := DEplot( MODEL, VARS, DOMAIN, RANGE, ICline, arrows=NONE, stepsize=0.1, linecolor=BLUE ):

> #linePLOT;

> display( [ dirPLOT, linePLOT ], title=`Direction Field & Straight-Line Solns` );

The fact that both approach the origin is consistent with the fact that the eigenvalues are negative. In addition, notice how the solutions in the second- and fourth-quadrants approach the origin more rapidly than those in the first- and third-quadrants. What explains this phenomenon?

>