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

Sketch the phase portrait for the system $\frac{\text{d}x}{\text{d}t}=3x-13y$, $\frac{\text{d}y}{\text{d}t}=5x+y$.

Find the general solution to the system of differential equations:

$\begin{cases} \frac{dx}{dt} = 2x + 3y \\ \frac{dy}{dt} = -6x - 9y \end{cases}$

Sketch the phase portrait for this system. Identify the nature of the equilibrium point at the origin and show the direction of trajectories.

Sketch the phase portrait for the system of differential equations: $\frac{dx}{dt} = 2x + 3y$ $\frac{dy}{dt} = -6x - 9y$

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

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

The phase portrait shows trajectories spiraling outward away from the origin. What can you conclude about the $\text{eigenvalues}$?

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

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.

The phase portrait for a linear system of differential equations shows trajectories spiraling inward toward the origin. Deduce the nature of the eigenvalues of the system matrix.

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 $A = \begin{pmatrix} 5 & 2 \\ 2 & 5 \end{pmatrix}$.

Trajectories of a linear system move away from the origin along the $x$-axis and towards the origin along the $y$-axis. State the signs of the eigenvalues and identify the type of equilibrium point at the origin.