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Question 1

Determine and sketch the phase portrait of the following system of linear differential equations: $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}-1 & -2 \\ 1 & -1\end{pmatrix}\mathbf{x}$

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Question 2

The question asks for the analysis of a $2 \times 2$ linear system of differential equations, including finding eigenvalues, eigenvectors, classifying the equilibrium, and sketching the phase portrait.

Determine and sketch the phase portrait of the system

$\frac{d\mathbf{x}}{dt} = \begin{pmatrix}-1 & 0 \\ 0 & -2 \end{pmatrix}\mathbf{x}$

Classify the equilibrium at the origin.

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Question 3

The question asks for the classification of a linear system's origin and a sketch of its phase portrait, requiring knowledge of eigenvalues, eigenvectors, and stability analysis.

Sketch the trajectories and classify the origin for the system $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}2 & 0 \\ 0 & 3\end{pmatrix}\mathbf{x}.$

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Question 4

Consider the system of differential equations given by $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}4 & -2 \\ 1 & 1\end{pmatrix}\mathbf{x}$

(a) Find the eigenvalues and corresponding eigenvectors for the system.

(b) Classify the equilibrium point at the origin.

(c) Sketch the phase portrait for the system, showing the eigenvectors and the direction of trajectories.

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Question 5

The system of linear differential equations is given by $\frac{d\mathbf{x}}{dt} = \mathbf{Ax}$ , where $\mathbf{A}$ is a constant $2 \times 2$ matrix.

Sketch the phase portrait for the system of differential equations $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}2 & 1 \\ 0 & 2\end{pmatrix}\mathbf{x}$ and classify the equilibrium at the origin.

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Question 6

Consider the linear system of differential equations $\frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x}$ , where $\mathbf{A} = \begin{pmatrix} 0 & 2 \\ -3 & 0 \end{pmatrix}$ and $\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$ .

1.

Find the eigenvalues of the matrix $\mathbf{A}$ .

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2.

Sketch the phase portrait for this system, clearly showing the direction of the trajectories.

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3.

Describe the motion of the system and state the nature and stability of the equilibrium point at the origin.

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Question 7

Systems of linear differential equations and phase portraits.

Sketch and classify the phase portrait of $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}1 & -4 \\ 1 & -1\end{pmatrix}\mathbf{x}.$

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Question 8

Classification of an equilibrium point and sketching of a phase portrait for a 2x2 linear system of differential equations given its general solution.

Given the general solution $\mathbf{x}(t)=C_1e^{3t}\begin{pmatrix}1\\2\end{pmatrix}+C_2e^{-t}\begin{pmatrix}2\\-1\end{pmatrix},$ sketch the phase portrait and classify the equilibrium point at the origin.

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Question 9

Systems of linear differential equations

Analyze and sketch the phase portrait of the system of differential equations given by:

$\frac{d\mathbf{x}}{dt} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\mathbf{x}$

Classify the equilibrium at the origin.

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Question 10

Sketch the trajectories for the planar system

\frac{d\mathbf{x}}{dt} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \mathbf{x}

and describe the motion.

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Question 11

Linear systems of differential equations, equilibrium points, and phase portraits.

Sketch and classify the equilibrium for the system $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}0 & 1 \\ -2 & 0.5\end{pmatrix}\mathbf{x}.$

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Question 12

Classify the equilibrium point at the origin and sketch the phase portrait for the system $\frac{\text{d}\mathbf{x}}{\text{d}t} = \begin{pmatrix} 5 & 3 \\ -3 & 5 \end{pmatrix} \mathbf{x}.$

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Question Type 4: Sketching trajectories of general solution of linear coupled differential equations onto a phase portrait Exercises