Question

Describe and graph trajectories of the given system. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} 9 & -3 \\ -1 & 11 \end{array}\right] \mathbf{y} $$

Short answer

**Question**: Describe and graph the trajectories of the system of linear differential equations represented by the matrix \(A = \begin{pmatrix} 9 & -3 \\ -1 & 11 \end{pmatrix}\). **Answer**: The general solution to the system of linear differential equations is: $$\mathbf{y}(t) = c_1 e^{8t}\begin{pmatrix} 3 \\ 1 \end{pmatrix} + c_2 e^{12t}\begin{pmatrix} 1 \\ -1 \end{pmatrix}$$ The trajectories of the system are expanding over time since both eigenvalues have positive real parts. To graph the trajectories, choose different initial conditions and plot the corresponding solutions (indicated by \(c_1\) and \(c_2\) in the general solution), showing how the system evolves over time.

Step-by-step solution

Convert matrix to a system of linear differential equations

From the given matrix, we can write the corresponding system of linear differential equations as follows: $$ \begin{cases} y_1' = 9y_1 - 3y_2 \\ y_2' =-y_1 + 11y_2 \end{cases} $$ where \(\mathbf{y} = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}\).

Find eigenvalues and eigenvectors of the matrix

For the given matrix \(A = \begin{pmatrix} 9 & -3 \\ -1 & 11 \end{pmatrix}\), we need to find its eigenvalues and eigenvectors. To find the eigenvalues, we solve for the determinant of \((A - \lambda I)\), where \(\lambda\) is the eigenvalue and \(I\) is the identity matrix. This gives us: $$\det(A - \lambda I) = \det\begin{pmatrix} 9-\lambda & -3 \\ -1 & 11-\lambda \end{pmatrix} = (9-\lambda)(11-\lambda) - (3)(1) = \lambda^2 - 20\lambda + 94$$ Solving for \(\lambda\), we obtain two eigenvalues, \(\lambda_1 = 8\) and \(\lambda_2 = 12\). Now, we find the eigenvectors for each eigenvalue by solving the equation \((A - \lambda I)\mathbf{v} = 0\), where \(\mathbf{v}\) is the eigenvector: For \(\lambda_1 = 8\), we have the system: $$\begin{cases} y_1 - 3y_2 = 0 \\ -y_1 + 3y_2 = 0 \end{cases}$$ The eigenvector corresponding to \(\lambda_1\) is \(\mathbf{v}_1 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\). For \(\lambda_2 = 12\), we have the system: $$\begin{cases} -3y_1 - 3y_2 = 0 \\ -y_1 - y_2 = 0 \end{cases}$$ The eigenvector corresponding to \(\lambda_2\) is \(\mathbf{v}_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}\).

Represent solutions in terms of eigenvalues and eigenvectors

The general solution to the system of linear differential equations is given by: $$\mathbf{y}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2$$ Plugging in our eigenvalues and eigenvectors, we have: $$\mathbf{y}(t) = c_1 e^{8t}\begin{pmatrix} 3 \\ 1 \end{pmatrix} + c_2 e^{12t}\begin{pmatrix} 1 \\ -1 \end{pmatrix}$$

Graph the trajectories of the system

The trajectories of the system can be represented by the general solution we derived in step 3: $$\mathbf{y}(t) = c_1 e^{8t}\begin{pmatrix} 3 \\ 1 \end{pmatrix} + c_2 e^{12t}\begin{pmatrix} 1 \\ -1 \end{pmatrix}$$ The qualitative behavior of the trajectories can be determined by the real parts of the eigenvalues, which are both positive. As a result, the trajectories will be expanding as time progresses. To graph the trajectories, you can choose different initial conditions and plot the corresponding solutions (indicated by \(c_1\) and \(c_2\) in the general solution), showing how the system evolves over time.

Key concepts

Eigenvalues

Eigenvalues provide important information about a matrix and are crucial in solving linear differential equations. Let's explore why they matter. In the context of the given system of differential equations, we find eigenvalues by solving the characteristic polynomial: \(\det(A - \lambda I) = \lambda^2 - 20\lambda + 94 = 0\). Finding the roots of this polynomial gives us the eigenvalues, \(\lambda_1 = 8\) and \(\lambda_2 = 12\). These numbers indicate how the solutions,over time, grow or decay. Since both eigenvalues are positive, any trajectory in the phase plane will tend to move away from the origin. Eigenvalues tell us about the system's stability and behavior, making them a fundamental concept in understanding differential equations.

Eigenvectors

Eigenvectors complement eigenvalues by indicating the direction in which certain transformations act. In the system we analyzed, eigenvectors are derived by solving \((A - \lambda I)\mathbf{v} = 0\). For each eigenvalue, there is a corresponding eigenvector. These vectors are significant because they show the paths along which a system evolves over time.

• For \(\lambda_1 = 8\), the eigenvector is \(\mathbf{v}_1 = \begin{pmatrix} 3 \ 1 \end{pmatrix}\), showing one of the stable directions.
• For \(\lambda_2 = 12\), the eigenvector is \(\mathbf{v}_2 = \begin{pmatrix} 1 \ -1 \end{pmatrix}\), indicating another path of expansion.

Recognizing these vectors helps us understand not only the rate of change, as indicated by eigenvalues, but also the specific paths that trajectories will follow in a phase space.

Trajectories

Trajectories describe how the system behaves over time, visualizing its evolution in the phase plane. In our example, they are derived from the general solution:\(\mathbf{y}(t) = c_1 e^{8t}\begin{pmatrix} 3 \ 1 \end{pmatrix} + c_2 e^{12t}\begin{pmatrix} 1 \ -1 \end{pmatrix}\). The terms \(c_1\) and \(c_2\) are constants determined by initial conditions. Because both eigenvalues are positive, all trajectories increase as time progresses, meaning the system is unstable.This understanding helps us predict future behavior of the system:

• If initial conditions are close to the positive eigenvector line; trajectories will tend to follow that direction initially more noticeably.
• With the rising exponential terms, trajectories rapidly diverge away from the origin.

By graphing these solutions, we gain a visual representation of the potential future states and how systems behave dynamically.

System of Differential Equations

Understanding a system of differential equations is key to modeling complex real-world phenomena. Our system, expressed as \(\begin{cases} y_1' = 9y_1 - 3y_2 \ y_2' = -y_1 + 11y_2 \end{cases}\), represents how two dependent variables, \(y_1\) and \(y_2\), change over time. Each differential equation relates the rate of change of one variable to both itself and possibly to the other.

The matrix form \([A \mathbf{y}]\) can concisely express more complicated systems by capturing interactions in a compact form, which facilitates finding solutions through techniques like matrix algebra and eigendecomposition. Effectively analyzing these systems requires:

• Representing them using matrices to simplify calculations.
• Identifying patterns and rates of growth, decay, or oscillation via eigenvalues and eigenvectors.

By converting the matrix to system forms and analyzing the solutions, we deepen our understanding of how interconnected variables evolve over time in various applications.