Understanding Linear Homogeneous Systems in the Plane

This video tutorial explores constant coefficient linear homogeneous systems in the plane, focusing on the behavior of solutions based on the eigenvalues of a 2x2 matrix. Six cases are discussed: real positive, real negative, mixed, purely imaginary, complex with positive real part, and complex with negative real part. Each case is illustrated with examples, showing how eigenvalues and eigenvectors influence the vector field and solution behavior, such as sources, sinks, saddle points, centers, spiral sources, and spiral sinks.

What is the primary characteristic of an autonomous system in the context of linear homogeneous systems?

• It depends on time.

• It is always unstable.

• It has no eigenvalues.

• It is independent of time.

How many cases of eigenvalue configurations are considered in the study of linear homogeneous systems?

• Four

• Five

• Six

• Seven

What is the behavior of solutions when both eigenvalues are real and positive?

• Solutions form a center.

• Solutions diverge from the origin.

• Solutions converge to the origin.

• Solutions form a saddle point.

What is the term used to describe the system when both eigenvalues are real and negative?

• Source

• Sink

• Saddle point

• Center

In the case of one positive and one negative eigenvalue, what type of point is formed?

• Saddle point

• Center

• Source

• Sink

What shape do the solutions form when the eigenvalues are purely imaginary?

• Ellipses

• Parabolas

• Hyperbolas

• Lines

What is the behavior of solutions when complex eigenvalues have a positive real part?

• Solutions shrink in magnitude.

• Solutions grow in magnitude.

• Solutions remain constant.

• Solutions form a saddle point.

What is the term for the system when complex eigenvalues have a negative real part?

• Spiral source

• Saddle point

• Spiral sink

• Center

Which of the following is NOT a characteristic of a spiral source?

• Solutions shrink in magnitude.

• Solutions grow in magnitude.

• Solutions spin around the origin.

• Solutions have a positive real part.

What happens to the solutions in a spiral sink?

• They remain constant.

• They grow in magnitude.

• They shrink in magnitude.

• They form a saddle point.