Two-dimensional systems and their vector fields

Classifying planar phase portraits by the signs and type of the eigenvalues: nodes, saddles, spirals, and centers, read straight off the vector field.

Notes on Diffy Qs: Differential Equations for Engineers, Two-dimensional systems and their vector fields

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2 exercises

Exercise 101

Describe the behavior of the following systems without solving:

$x' = x + y$ , $y' = x-y$ .
$x_1' = x_1 + x_2$ , $x_2' = 2 x_2$ .
$x_1' = -2x_2$ , $x_2' = 2 x_1$ .
$x' = x + 3y$ , $y' = -2x-4y$ .
$x' = x - 4y$ , $y' = -4x+y$ .

2.5

Exercise 102

Suppose that $\vec{x}\,' = P \vec{x}$ where $P$ is a 2 by 2 matrix with eigenvalues $2\pm i$ . Describe the behavior.

2.5