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In each exercise, the general solution of a \((2 \times 2)\) linear system \(\mathbf{y}^{\prime}=A \mathbf{y}\) is given, along with an initial condition. Sketch the phase plane solution trajectory that satisfies the given initial condition. $$ \mathbf{y}(t)=c_{1} e^{2 t}\left[\begin{array}{l} 1 \\ 1 \end{array}\right]+c_{2} e^{t}\left[\begin{array}{l} 0 \\ 1 \end{array}\right], \quad \mathbf{y}(0)=\left[\begin{array}{r} 0 \\ -2 \end{array}\right] $$

Short Answer

Expert verified
Question: Sketch the phase plane solution trajectory of the 2x2 linear system with the initial condition (0, -2) given the general solution in the form of distinct eigenvalues and eigenvectors. Solution: The trajectory will follow the vertical line \(x = 0\) and moves downward in the \(y\) direction, based on the given initial condition of (0, -2). The equilibrium point (0,0) acts as a "source" from which trajectories move away.

Step by step solution

01

Determine the constants \(c_1\) and \(c_2\) using the initial condition

Substitute the initial condition into the general solution: $$ \mathbf{y}(0)=c_{1} e^{2 (0)}\left[\begin{array}{l} 1 \\\ 1 \end{array}\right]+c_{2} e^{(0)}\left[\begin{array}{l} 0 \\\ 1 \end{array}\right]=\left[\begin{array}{r} 0 \\\ -2 \end{array}\right] $$ This equation simplifies to: $$ \left[\begin{array}{l} c_{1} \\ c_{1} + c_{2} \end{array}\right]=\left[\begin{array}{r} 0 \\\ -2 \end{array}\right] $$ Solving this system of linear equations, we get \(c_1 = 0\) and \(c_2 = -2\).
02

Analyze the eigenspaces corresponding to the eigenvalues

The general solution is given in the form that corresponds to two real and distinct eigenvalues \(e^{2t}\) and \(e^t\) with their corresponding eigenvectors \(\left[\begin{array}{l} 1 \\ 1 \end{array}\right]\) and \(\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\) respectively. The eigenspace corresponding to \(e^{2t}\) is spanned by \(\left[\begin{array}{l} 1 \\ 1 \end{array}\right]\), which represents a line \(y=x\). The eigenspace corresponding to \(e^t\) is spanned by \(\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\), which represents a vertical line \(x = 0\).
03

Sketch the phase plane trajectory

With the obtained constants \(c_1\) and \(c_2\), our specific solution becomes: $$ \mathbf{y}(t)=-2 e^{t}\left[\begin{array}{l} 0 \\\ 1 \end{array}\right] =\left[\begin{array}{l} 0 \\\ -2 e^{t} \end{array}\right] $$ All solutions to the system move away from the equilibrium point at \((0,0)\) along the vertical line \(x = 0\). And because of the negative coefficient \(-2\) for the eigenvector \(\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\), the solution trajectory will follow the vertical line, moving downward in the \(y\) direction due to the \(-2 e^{t}\) factor. Based on this information, sketch the phase plane with the solution trajectory going downwards along the vertical line \(x = 0\) and starting from the initial point \(\left[\begin{array}{r} 0 \\ -2 \end{array}\right]\). The equilibrium point \((0,0)\) should be marked as a "source" (a point from which the trajectories move away).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phase Plane Trajectories
Phase plane trajectories are a useful way to visualize the behavior of solutions to a system of differential equations over time. In a two-dimensional linear system, the phase plane is a graphical representation where each point corresponds to a state of the system. The axes represent variables such as position and velocity, or other quantities of interest. Trajectories within this plane demonstrate how the system evolves, starting from initial conditions.
For the provided linear system, the trajectory can be plotted using the given initial condition. Here, we observe that the trajectory spans a vertical line along the x-axis due to the structure of the solution. Such trajectories offer insight into stability and behavior of dynamic systems, showcasing how solutions can diverge or converge around equilibrium points.
Eigenvalues and Eigenvectors
Understanding eigenvalues and eigenvectors is crucial in the analysis of linear differential systems. They provide insight into the dynamics of the system, indicating the direction and nature of the trajectories in the phase plane. Eigenvalues are scalars that determine whether trajectories grow, shrink, or oscillate.
In our example exercise, we have two distinct eigenvalues: one corresponding to an exponential term with value 2, and another with value 1. Their respective eigenvectors define the directions in which these exponential changes occur. An eigenvector is essentially a direction in which the transformation acts merely by scaling, not rotating, the vector. For our system, the eigenvectors \([1, 1]\) and \([0, 1]\) indicate that solutions spread along the lines they represent.
  • Real and positive eigenvalues typically result in trajectories moving away from the origin, indicating instability.
  • The magnitude of eigenvalues suggests the rate of divergence or convergence.
Initial Conditions
Initial conditions define the starting point of a trajectory in the phase plane. They are imperative for determining a specific solution from a general solution to a differential equation. By substituting initial conditions into the general solution, we can solve for unknown constants, tailoring the solution to a particular scenario or state.
In the given system, the initial condition was \(\ \mathbf{y}(0) = [0, -2] \ \). This condition was used to find constants \(\ c_1 \) and \(\ c_2 \), resulting in a specific trajectory starting precisely at the point \(\ (0, -2) \) in the phase plane. The initial conditions determine not only where but also how the system begins to evolve over time, influencing the overall path it takes.
Differential Equations Solutions
Solving differential equations involves finding expressions that describe the behavior of dynamic systems. Linear differential equations are a class where solutions consist of particular forms influenced by the system's properties, like eigenvalues and eigenvectors.
For linear systems, such as the one discussed, the general solution comprises terms with exponential growth or decay, dictated by the eigenvalues. In our exercise, this solution structure is crafted by combining solutions associated with each eigenvalue and its respective eigenvector. The constants associated with these solutions are determined by initial conditions to provide a unique solution trajectory.
  • Linear systems allow for superposition, whereby individual solutions from several terms combine to form the general solution.
  • The exponential terms in solutions suggest growth patterns and help forecast future states of the system.
Solving such systems helps predict how the system responds to various initial conditions, thereby constructing a comprehensive picture of its dynamic behavior.

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Most popular questions from this chapter

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