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Exercises for Question Type 6: Determining the type of eigenvalue given a phase portrait diagram - IB | RevisionDojo

Question 1

The following question relates to the analysis of linear systems of differential equations in the form $\frac{\text{d}\mathbf{x}}{\text{d}t} = \mathbf{Ax}$ , where $\mathbf{x} \in \mathbb{R}^2$ and $\mathbf{A}$ is a $2 \times 2$ real constant matrix.

A linear system of differential equations is represented by the phase portrait shown below:

Identify the type of equilibrium point at the origin and characterize the nature of the eigenvalues of the system matrix $\mathbf{A}$ .

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Question 2

This question involves the analysis of a phase portrait for a system of linear differential equations.

The following phase portrait shows the trajectories of a system of linear differential equations $\frac{\text{d}\mathbf{x}}{\text{d}t} = \mathbf{A}\mathbf{x}$ near the origin.

Phase portrait of a degenerate stable node

Identify the nature of the eigenvalues of matrix $\mathbf{A}$ and the type of equilibrium point at the origin. Justify your answers.

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Question 3

This question tests the ability to qualitatively analyze a phase portrait of a system of linear differential equations $\frac{\text{d}\mathbf{x}}{\text{d}t} = \mathbf{Ax}$ to identify the classification of an equilibrium point and the nature of the associated eigenvalues and eigenvectors.

The following phase portrait shows the trajectories of a system of linear differential equations $\frac{\text{d}\mathbf{x}}{\text{d}t} = \mathbf{Ax}$ near an equilibrium point at the origin.

Phase portrait of an improper unstable node

Classify the equilibrium point and describe the nature of the eigenvalues and eigenvectors of matrix $\mathbf{A}$ .

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Question 4

The following phase portrait represents the behavior of a linear system $\dot{\mathbf{x}} = A\mathbf{x}$ in the plane.

Examine the following phase portrait:

Phase portrait showing trajectories spiralling inward towards the origin

Determine the equilibrium type at the origin and characterize the eigenvalues of the corresponding system.

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Question 5

Identify the type of equilibrium point and describe its eigenvalues based on the given phase portrait.

Consider the following phase portrait for a linear system of differential equations:

Phase portrait showing trajectories spiraling outward from the origin

Identify the equilibrium type at the origin and describe the nature of the eigenvalues, $\lambda$ , associated with this system.

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Question 6

Consider the following phase portrait for a system of linear differential equations:

Phase portrait showing trajectories near the origin

Classify the equilibrium point at the origin and describe the nature of its corresponding eigenvalues.

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Question 7

Examine the following phase portrait:

Phase portrait showing an unstable spiral at the origin

Determine the type of equilibrium point at the origin and describe the characteristics of its eigenvalues.

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Question 8

The following phase portrait is given for a system of linear differential equations:

Phase portrait of an equilibrium point at the origin

Classify the equilibrium at the origin and state the nature of its eigenvalues.

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Question 9

Determine the type of equilibrium and the nature of the eigenvalues from a given phase portrait.

Consider the phase portrait shown below:

Determine the type of equilibrium at the origin and describe the eigenvalues (real or complex, distinct or repeated, and sign).

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Question 10

Level: HL Paper: 2

Analyze the phase portrait shown below for a system of linear differential equations:

Identify the type of equilibrium point at the origin and describe the nature of its eigenvalues.

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Question 11

The diagram shows the phase portrait of a linear system of differential equations near the origin.

Phase portrait showing closed circular orbits around the origin

Identify the type of equilibrium point at the origin and describe the nature of its eigenvalues.

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Question Type 6: Determining the type of eigenvalue given a phase portrait diagram Exercises