Question

Plot trajectories of the given system. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} -2 & -3 \\ 3 & 4 \end{array}\right] \mathbf{y} $$

Short answer

Answer: The four main steps involved in plotting the trajectories are: 1. Finding the eigenvalues of the matrix by computing the characteristic equation and solving for the eigenvalues. 2. Calculating the eigenvectors corresponding to each eigenvalue using the equation (A - λI)v = 0. 3. Writing the general solution as a linear combination of the eigenvectors, where each component is multiplied by the exponential of the eigenvalue times the scalar variable t. 4. Plotting the trajectories of the system using the general solution by choosing multiple values for the scalar constants and graphing the changing components of the solution.

Step-by-step solution

Finding the eigenvalues of the matrix

To find the eigenvalues, we will first compute the characteristic equation, a quadratic equation in terms of the trace, determinant, and eigenvalue(s) of the matrix. Use the following formula: $$ \text{det} (A - \lambda I) = 0 $$ where \(A\) is the matrix, \(\lambda\) is the eigenvalue, \(I\) is the identity matrix, and \(\text{det}\) represents the determinant. In our case: $$ \text{det} \left(\begin{array}{rr} -2-\lambda & -3 \\ 3 & 4-\lambda \end{array}\right) = 0 $$ Calculate the determinant and solve for the eigenvalues, \(\lambda\).

Calculate the eigenvectors for each eigenvalue

For each of the eigenvalues found in Step 1, we will now calculate the respective eigenvectors. An eigenvector \(\mathbf{v}\) is a non-zero vector that satisfies the following equation: $$ (A-\lambda I)\mathbf{v} = 0 $$ Substitute each of the eigenvalues into this equation, and solve for the eigenvectors.

Write the general solution

Now that we have the eigenvalues and eigenvectors, we will combine them into the general solution for the given system. The general solution is a linear combination of the eigenvectors, where each component is multiplied by the exponential of the eigenvalue times the scalar variable t. The general solution takes the following form: $$ \mathbf{y}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2 $$ where \(c_1\) and \(c_2\) are arbitrary constants, \(\lambda_1\) and \(\lambda_2\) are the eigenvalues, and \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are the corresponding eigenvectors. Substitute the specific eigenvalues and eigenvectors into this equation.

Plot the trajectories

Finally, we will plot the trajectories of our system using the general solution. Choose multiple values of the scalar constants \(c_1\) and \(c_2\), and see how the components of the solution change. Plot these trajectories on a graph for visualization. By following these four steps, the trajectories of the given system have been plotted.

Key concepts

Eigenvalues Calculation

Understanding the eigenvalues of a matrix is a fundamental step in analyzing the behavior of a system of differential equations. To illustrate the calculation of eigenvalues, let's consider our matrix from the exercise:

\begin{align*}A &= \begin{bmatrix} -2 & -3 \ 3 & 4d{bmatrix}.\end{align*}
Eigenvalues are special scalars associated with a linear system that, when multiplied by an identity matrix and subtracted from the original matrix, render it singular. The characteristic equation needed here is represented by \( \text{det} (A - \lambda I) = 0 \), where \(I\) stands for the identity matrix, and \(\lambda\) is the eigenvalue we wish to find. By calculating the determinant, we end up with a polynomial equation in terms of \(\lambda\). Solving for \(\lambda\) yields the eigenvalues of the matrix, which tell us about the stability and types of solutions we can expect from the system. Larger eigenvalues correspond to trajectories that diverge more quickly, whereas smaller ones correspond to converging or slower diverging trajectories.

• Calculate the determinant of \(A - \lambda I\).
• Solve the characteristic polynomial for \(\lambda\).

Eigenvalues are the roots of this polynomial equation, and they play a critical role in defining the trajectories of the system.

Eigenvectors Computation

Once the eigenvalues have been identified, the next step is to find the respective eigenvectors. These are non-zero vectors that, when transformed by the matrix, result in a vector that is a scalar multiple of the original one. The computation involves solving the equation:\

\[ (A - \lambda I)\mathbf{v} = 0 \]
for each eigenvalue \(\lambda\). Here's how to calculate eigenvectors step by step:

• For each eigenvalue, subtract \(\lambda\) times the identity matrix from the matrix \(A\).
• Solve the resulting homogeneous system to find the non-zero vector \(\mathbf{v}\).

Eigenvectors provide the directions along which the system's trajectories move. In differential equations, each eigenvector associated with a real eigenvalue indicates a principal direction of motion for the system.

General Solution of System

With eigenvectors and eigenvalues in hand, we can construct the general solution of the system. The general solution will take the form of a linear combination of exponential functions of time \(t\) scaled by the eigenvalues and directed by the eigenvectors.

The general solution for a 2x2 system is expressed as:\

\[ \mathbf{y}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2 \]
where \(c_1\) and \(c_2\) are arbitrary constants that would be determined by initial conditions, and \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are the eigenvectors corresponding to eigenvalues \(\lambda_1\) and \(\lambda_2\), respectively.

As time grows, the terms involving the exponential functions signify how the system evolves. If the eigenvalues are real and distinct, the solution will exhibit typical exponential growth or decay, with each term in the solution growing or shrinking independently along the direction of its corresponding eigenvector.

Plotting Trajectories

Plotting trajectories of a system of differential equations provides powerful visual insights into the behavior over time. For the system given by \( \mathbf{y}' = A\mathbf{y} \), trajectories can be plotted using the general solution. Once we identify various pairs of constants \(c_1\) and \(c_2\), we generate distinct solutions that correspond to different initial conditions. By computing the values of \(\mathbf{y}(t)\) over a range of \(t\) values, and plotting these points, we visualize the flow of the trajectories in the system's phase space.

Here are the steps for plotting:

• Select multiple pairs of \(c_1\) and \(c_2\) to represent different starting positions.
• Calculate the evolution of the system over time using the general solution for each pair.
• Plot these solutions on a graph, typically with the x-axis and y-axis representing the elements of vector \(\mathbf{y}(t)\).

As a result, the graph showcases the paths taken by the system from various starting points, illustrating the influence of the system's eigenvalues and eigenvectors on its long-term behavior.