Question

In each of Problems 1 through 6 find the general solution of the given system of equations and describe the behavior of the solution as \(t \rightarrow \infty\). Also draw a direction field and plot a few trajectories of the system. $$ \mathbf{x}^{\prime}=\left(\begin{array}{ll}{3} & {-2} \\ {2} & {-2}\end{array}\right) \mathbf{x} $$

Short answer

Based on the given step-by-step solution, the general solution for the given linear system of differential equations is: $$ \mathbf{x}(t) = c_1 e^{t}\left(\begin{array}{c}{1} \\\ {1}\end{array}\right) + c_2 e^{2t}\left(\begin{array}{c}{2} \\\ {1}\end{array}\right) $$ As \(t\) approaches infinity, the solution will approach a straight line in the direction of the eigenvector \(\mathbf{v}_2 = \begin{pmatrix} 2\\ 1 \end{pmatrix}\). A direction field can be plotted using various tools, and trajectories of the system can be found by choosing different values for \(c_1\) and \(c_2\) in the general solution.

Step-by-step solution

Find the eigenvalues and eigenvectors of the coefficient matrix

To find the eigenvalues of the given matrix, we need to solve the following characteristic equation: $$ \text{det}\left(\begin{array}{cc}{3 - \lambda} & {-2} \\\ {2} & {-2-\lambda}\end{array}\right) = 0 $$ Calculating the determinant, we get: $$ (3 - \lambda)(-2 - \lambda) - (-2)(2)=0 $$ Expanding and solving the quadratic equation, we find that the eigenvalues are \(\lambda_1 = 1\) and \(\lambda_2 = 2\). Now, we will find the eigenvectors corresponding to each eigenvalue. For \(\lambda_1=1\), we solve the following system of linear equations: $$ \left(\begin{array}{cc}{2} & {-2} \\\ {2} & {-3}\end{array}\right)\left(\begin{array}{c}{x_1} \\\ {x_2}\end{array}\right) = 0 $$ Solving the system, we find that the eigenvector corresponding to \(\lambda_1=1\) is $$ \mathbf{v}_{1}=\left(\begin{array}{c}{1} \\\ {1}\end{array}\right) $$ For \(\lambda_2=2\), we solve the following system of linear equations: $$ \left(\begin{array}{cc}{1} & {-2} \\\ {2} & {-4}\end{array}\right)\left(\begin{array}{c}{x_1} \\\ {x_2}\end{array}\right) = 0 $$ Solving the system, we find that the eigenvector corresponding to \(\lambda_2=2\) is $$ \mathbf{v}_{2}=\left(\begin{array}{c}{2} \\\ {1}\end{array}\right) $$

Find the general solution of the equivalent system of uncoupled equations

Using the eigenvalues and eigenvectors found, we can write the general solution of the system as: $$ \mathbf{x}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2 = c_1 e^{t}\left(\begin{array}{c}{1} \\\ {1}\end{array}\right) + c_2 e^{2t}\left(\begin{array}{c}{2} \\\ {1}\end{array}\right) $$ where \(c_1\) and \(c_2\) are constants depending on the initial conditions of the problem.

Describe the behavior of the solution as \(t \rightarrow \infty\)

As t approaches infinity, the term \(e^{2t}\) will dominate over \(e^t\), and the solution will approach the eigenvector \(\mathbf{v}_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\). Therefore, the overall behavior of the solution as \(t\rightarrow \infty\) will be determined by the eigenvector \(\mathbf{v}_2\), and the solutions will approach a straight line in the direction of the eigenvector \(\mathbf{v}_2\).

Draw a direction field and plot a few trajectories

To draw a direction field, you can use software like MATLAB, Python, or an online tool like Desmos. To do so, plot arrows or short line segments indicating the direction of the flow at each point in the xy-plane. To plot a few trajectories, you can use the general solution found in Step 2 and choose different values for \(c_1\) and \(c_2\). Then, integrate the trajectories in the direction field starting from different initial conditions. The trajectories should give a visual representation of the behavior as \(t \rightarrow \infty\), converging to a straight line in the direction of the eigenvector \(\mathbf{v}_2\).

Key concepts

Eigenvectors

In a system of differential equations, eigenvectors play a crucial role. They are specific, non-zero vectors that, when multiplied by a given matrix, result in a scalar multiple of themselves. Essentially, they tell us the direction in which a linear transformation stretches or compresses space.

To find an eigenvector, you solve the equation of the form \[(A - \lambda I)\mathbf{v} = 0\]where \(A\) is the coefficient matrix of the system, \(\lambda\) represents an eigenvalue, \(I\) is the identity matrix of the same size as \(A\), and \(\mathbf{v}\) is the eigenvector.

In our problems, solving these equations gives us the vectors \(\mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix}\) and \(\mathbf{v}_2 = \begin{pmatrix} 2 \ 1 \end{pmatrix}\), which correspond to their respective eigenvalues. These directions are vital as they indicate the behavior of the entire system's solutions over time.

Eigenvalues

Eigenvalues are really helpful in understanding how a system of differential equations behaves. They are scalar values that, in combination with eigenvectors, describe the action of a matrix on a vector.

To find the eigenvalues, you form the characteristic polynomial \[det(A - \lambda I) = 0\] and solve it for \(\lambda\).

In this exercise, we found the eigenvalues to be \(\lambda_1 = 1\) and \(\lambda_2 = 2\). These values tell us how the vectors stretch or rotate. Each eigenvalue relates to an eigenvector, defining how the matrix transforms the system state in that specific direction.

As time progresses in dynamic systems, eigenvalues help predict whether these values will lead towards infinity or decay to zero.

General Solution

The general solution of a system of differential equations offers a comprehensive overview of all possible behaviors of the system. It is composed using the eigenvectors and eigenvalues.

For our specific system, the general solution is expressed as: \[\mathbf{x}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2\] where \(c_1\) and \(c_2\) are arbitrary constants determined by initial conditions. The expression combines each eigenvector multiplied by an exponential term determined by its eigenvalue.

This solution captures both immediate behavior and long-term trends as time \(t\) approaches infinity. In our case, because \(\lambda_2 = 2\) is greater, its corresponding term grows more rapidly, thus dominating eventual system behavior.

Direction Field

Direction fields are a visual tool for understanding how differential equations behave across different initial conditions and states. They provide a "field" of arrows or lines showing the direction of the system's solutions at different points in space.

When plotting a direction field, each point on the plane has a short line segment pointing in the direction determined by the differential equation at that point.

For our exercise, using software like Python or MATLAB, you can plot the system's vectors indicating the flow's directions. It beautifully represents how the system evolves and helps visualize trajectory convergence towards eigenvectors for \(t \rightarrow \infty\). When you draw the direction field for this exercise, you'll notice that trajectories start diverging or converging, influenced heavily by the eigenvector \(\mathbf{v}_2 = \begin{pmatrix} 2 \ 1 \end{pmatrix}\), since it dominates as time progresses.