Question Type 3: Finding the general solution to linear coupled differential equations with any type of eigenvalues Exercises

Trajectories move away from the origin along the x-axis and towards the origin along the y-axis. What are the signs of the eigenvalues, and what type of equilibrium is at the origin?

The phase portrait shows trajectories spiraling outward away from the origin. What can you conclude about the eigenvalues?

The phase portrait shows trajectories spiraling inward toward the origin. What can you conclude about the eigenvalues?

Sketch the phase portrait for the system $\frac{dx}{dt}=5x+2y$, $\frac{dy}{dt}=2x+2y$.

Sketch the phase portrait for the system $\frac{dx}{dt}=2x+3y$, $\frac{dy}{dt}=-6x-9y$.

Determine whether the origin is a stable node, unstable node, stable focus, unstable focus, or saddle for the system $\mathbf{x}'=A\mathbf{x}$ with $A=\begin{pmatrix}5&2\\2&5\end{pmatrix}$.

Sketch the phase portrait for the system $\frac{dx}{dt}=3x-13y$, $\frac{dy}{dt}=5x+y$.

Find the general solution to the system of differential equations $\frac{dx}{dt}=5x+2y$, $\frac{dy}{dt}=2x+2y$.

For the system $\mathbf{x}'=A\mathbf{x}$ with $A=\begin{pmatrix}0&1\\-2&-3\end{pmatrix}$, find the general solution and classify the origin.

For the matrix $A=\begin{pmatrix}-3&-4\\1&-1\end{pmatrix}$, find its eigenvalues and eigenvectors, then classify the equilibrium at the origin for $\mathbf{x}'=A\mathbf{x}$.

Find the general solution to the system $\frac{dx}{dt}=3x-13y$, $\frac{dy}{dt}=5x+y$.

Find the general solution to the system $\frac{dx}{dt}=2x+3y$, $\frac{dy}{dt}=-6x-9y$.