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

Determine and sketch the phase portrait of the system rac{d\mathbf{x}}{dt} = \begin{pmatrix}-1 & 0 \\ 0 & -2 \end{pmatrix}\mathbf{x} Classify the equilibrium at the origin.

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

Analyze and sketch the phase portrait of $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\mathbf{x}.$ Classify the equilibrium at the origin.

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.

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.

Analyze the system $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}4 & -2 \\ 1 & 1\end{pmatrix}\mathbf{x}$ by finding its eigenvalues, eigenvectors, classifying the origin, and sketching the phase portrait.

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

Determine and sketch the phase portrait of $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}-1 & -2 \\ 1 & -1\end{pmatrix}\mathbf{x}.$

Sketch the phase portrait of $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}0 & 2 \\ -3 & 0\end{pmatrix}\mathbf{x}$ and describe the motion.

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

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

Classify and sketch the trajectories for the system $\frac{d\mathbf{x}}{dt} = \begin{pmatrix}5 & 3 \\ -3 & 5\end{pmatrix}\mathbf{x}.$