Question

The matrices of the systems in exercises are singular. Describe and graph the trajectories of nonconstant solutions of the given systems. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} 1 & -2 \\ -1 & 2 \end{array}\right] \mathbf{y} $$

Short answer

Based on the provided step by step solution, please answer the following question: **Question:** Describe the general behavior of the trajectories of nonconstant solutions for the given singular matrix differential equation. **Answer:** The general behavior of the trajectories of nonconstant solutions is exponential growth over time, as both eigenvalues are positive. Trajectories might approach the x-axis or y-axis depending on the initial conditions (i.e., the values of \(c_1\) and \(c_2\)). They do not converge to a fixed point as the matrix is singular.

Step-by-step solution

Find eigenvalues and eigenvectors for the matrix

To find the eigenvalues and eigenvectors for the given matrix, we need to set up and solve the characteristic equation \(\det(\mathbf{A} - \lambda \mathbf{I}) = 0\), where \(\mathbf{A}\) is the given matrix and \(\mathbf{I}\) is the identity matrix. This will help us find the general solution of the given system. The matrix equation is given by: $$ \mathbf{A}= \left[\begin{array}{rr} 1 & -2 \\ -1 & 2 \end{array}\right] $$ Subtract \(\lambda\mathbf{I}\) from \(\mathbf{A}\) and set the determinant to zero:$$ \det(\mathbf{A} - \lambda \mathbf{I}) = \det\left(\left[\begin{array}{cc} 1-\lambda & -2 \\ -1 & 2-\lambda \end{array}\right]\right) = (1-\lambda)(2-\lambda)-(-2)(-1) = 0 $$Solve the above equation for eigenvalues.

Solve for eigenvalues

Solve the quadratic equation \((1-\lambda)(2-\lambda)-2 = 0\) for \(\lambda\). The roots of the given equation are: \(\lambda_1 = 1\) and \(\lambda_2 = 2\).

Solve for eigenvectors

Now, let's find the eigenvectors corresponding to each eigenvalue by solving the equation \((\mathbf{A} - \lambda \mathbf{I})\mathbf{v} = \mathbf{0}\) for each eigenvalue. For \(\lambda_1 = 1\), the equation becomes:$$ \left[\begin{array}{cc} 0 & -2 \\ -1 & 1 \end{array}\right]\left[\begin{array}{c} v_{11} \\ v_{12} \end{array}\right] = \mathbf{0} $$Solve the above system for \(\mathbf{v}_1\), we obtain: $$ \mathbf{v}_1 = \left[\begin{array}{c} 1 \\ \dfrac{1}{2} \end{array}\right] $$ For \(\lambda_2 = 2\), the equation becomes:$$ \left[\begin{array}{cc} -1 & -2 \\ -1 & 0 \end{array}\right]\left[\begin{array}{c} v_{21} \\ v_{22} \end{array}\right] = \mathbf{0} $$Solve the above system for \(\mathbf{v}_2\), we get: $$ \mathbf{v}_2 = \left[\begin{array}{c} 1 \\ -\dfrac{1}{2} \end{array}\right] $$

Write the general solution using eigenvalues and eigenvectors

Now, we have the eigenvalues and their corresponding eigenvectors, so we can write the general solution of the given system as:$$ \mathbf{y}(t) = c_1 \mathrm{e}^{\lambda_1 t} \mathbf{v}_1 + c_2 \mathrm{e}^{\lambda_2 t} \mathbf{v}_2 = c_1 \mathrm{e}^t \left[\begin{array}{c} 1 \\ \dfrac{1}{2} \end{array}\right] + c_2 \mathrm{e}^{2t} \left[\begin{array}{c} 1 \\ -\dfrac{1}{2} \end{array}\right] $$Here, \(c_1\) and \(c_2\) are arbitrary constants.

Analyze and describe the trajectories of nonconstant solutions

The general solution exhibits trajectories that change over time. Since the system's matrix is singular, it implies that the solutions will not converge to a fixed point. Also, both eigenvalues are positive, which means that the nonconstant solutions will grow exponentially in time and not oscillate. The trajectories of the system will be of the form:$$ \mathbf{y}(t) = \left[\begin{array}{c} c_1 \mathrm{e}^t + c_2 \mathrm{e}^{2t} \\ \dfrac{1}{2}c_1 \mathrm{e}^t - \dfrac{1}{2}c_2 \mathrm{e}^{2t} \end{array}\right] $$In summary, the trajectories will have an exponential growth over time. They might approach the x-axis or the y-axis depending on the initial conditions (i.e., the values of \(c_1\) and \(c_2\)).

Graph the trajectories

To graph the trajectories of nonconstant solutions, we can sketch a few representative trajectories by choosing various combinations of \(c_1\) and \(c_2\). It is important to highlight the overall behavior of solutions. Keep in mind that various initial conditions will result in different trajectories. Some trajectories may move away from the origin in the direction of eigenvectors, while others may not. They might approach the x-axis or y-axis depending on the values of \(c_1\) and \(c_2\). It is important to sketch a few sample trajectories to capture the overall behavior of the solutions.

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is crucial for analyzing systems of differential equations. These elements help determine the system's behavior over time.
- **Eigenvalues** are numbers that provide information about the growth or decay rate of solutions. When applied to a given matrix, they indicate the presence of solutions that grow, decay, or oscillate.
- **Eigenvectors**, on the other hand, are vectors that do not change direction under the transformation described by the matrix. They give information about the direction of growth or decay for solutions.
In the provided exercise, by solving the characteristic equation, we found the eigenvalues \( \lambda_1 = 1\) and \( \lambda_2 = 2\). Both eigenvalues are positive, meaning solutions involving these will grow exponentially over time instead of oscillating or converging to a fixed point. Corresponding to each eigenvalue, we also determined the eigenvectors needed to describe the direction of these growing solutions.

Singular Matrix

A matrix is called singular if its determinant is zero. This attribute has implications for the system of differential equations it describes.
- **Characteristic of a Singular Matrix:** In this context, singular matrices do not possess an inverse, which reflects some limitations in solving equations using such matrices.
In our specific example, the matrix \[\begin{pmatrix} 1 & -2 \ -1 & 2 \end{pmatrix}\] is singular, as its determinant equals zero. Although finding the determinant might initially appear challenging, remember that a singular matrix often describes systems whose solutions do not converge to a single fixed point. Instead, it indicates that the trajectories of solutions diverge or evolve over time. This means solutions will often exhibit exponential growth, as demonstrated by the eigenvalues previously found.

Trajectories of Solutions

The solutions to systems of differential equations are often represented as trajectories on a graph. These trajectories give visual insight into how solutions behave over time.
- **Behavioral Interpretation:** For this exercise, with both eigenvalues positive, trajectories do not converge or oscillate. They diverge, with solutions exponentially growing over time.
Consider the general solution: \[\mathbf{y}(t) = \left[ \begin{array}{c} c_1 e^t + c_2 e^{2t} \ \frac{1}{2}c_1 e^t - \frac{1}{2}c_2 e^{2t} \end{array} \right]\] where \( c_1 \) and \( c_2 \) are constants based on initial conditions. Trajectories will expand depending on the interaction of these constants.
The system's singular matrix and positive eigenvalues assure us that the trajectories take shapes indicative of growth, with pathways moving increasingly away from the origin as time progresses. Understanding these trajectories facilitates insights into the long-term behavior of the system.

Systems of Differential Equations

A system of differential equations involves multiple interconnected equations representing a set of variables' rates of change relative to each other. Such systems are vital in modeling phenomena where two or more quantities evolve together over time.
- **Analysis Methods:** Methods to solve include finding eigenvalues and eigenvectors of the matrix associated with the system. These mathematical tools simplify analyzing how each component of the system changes in unison.
In our exercise, analyzing the system of equations led us to comprehend the system’s evolution through its eigenvalues and eigenvectors. This understanding allows us to predict how trajectories would develop when different initial conditions are set.
Understanding systems of differential equations leads to insights into complex systems, aiding predictions about real-world phenomena such as population dynamics, chemical reactions, or mechanical systems.