Trajectories of a linear system first order diff. equations

[Nikitin]

In linear algebra, you can have systems of differential equations represented by matrices.

What does a "trajectory graph" of such a system show, exactly? And how can you draw one?

What's the difference between such a trajectory-graph and an ordinary slope-field for a single linear differential equation?

Thanks! :)

 

[HallsofIvy]

The "slope field" is a graph showing short straight lines at each (x, y) point (well, in reality as many as possible without one covering another!), showing the slope of a solution to the differential equation. A "trajectory graph" is a graph showing an number of actuals "trajectories" (solutions) of the differential equation, with different initial conditions as possible.

If you have a "slope field" for a differential equation, each of the trajectories must be parallel to the slope field lines at each point.