Thread: Linear System: Phase Planes

Hi Everyone,

The system given is:



The eigenvalues and the corresponding eigenvectors I found were:





Therefore,




Now the text says to sketch several trajectories in the phase plane and also sketch some typical graphs of versus t .

The text does not really explain how its generating the example graphs, it just shows pictures of general graphs. How would I go about sketching these? Without using a computer of course.

Originally Posted by utopiaNow

Hi Everyone,

The system given is:



The eigenvalues and the corresponding eigenvectors I found were:





Therefore,




Now the text says to sketch several trajectories in the phase plane and also sketch some typical graphs of versus t .

The text does not really explain how its generating the example graphs, it just shows pictures of general graphs. How would I go about sketching these? Without using a computer of course.

First graph the two lines given by the eigenvectors: <1, 3> lies along the line y= 3x and <1, 1> along the line y= x. Since both eigenvalues are positive, the "flow" is outward. All other solutions will be outward along curves emanating from the origin curving toward y= x because it corresponds to the larger eigenvalue, 4 so the flow is "faster" there.