Problem 20 - Advanced Engineering Mathematics

Question

In Problems, for the given linear dynamical system (taken from Exercises 10.2) (a) find the general solution and determine whether there are periodic solutions, (b) find the solution satisfying the given initial condition, and, (c) with the aid of a graphing utility, plot the solution in part (b) and indicate the direction in which the curve is traversed. $$ \begin{aligned} &x^{\prime}=x+y \\ &y^{\prime}=-2 x-y, \mathbf{X}(0)=(-2,2)(\text { Problem } 36, \text { Exercises } 10.2) \end{aligned} $$

Step-by-Step Solution

Answer

The general solution is periodic with period \( 2\pi \). The solution with initial condition \( \mathbf{X}(0) = (-2, 2) \) is an ellipse.

Step 1: Write the System in Matrix Form

The given system of differential equations is \( x' = x + y \) and \( y' = -2x - y \). We write this system in matrix form as \( \mathbf{X}' = A\mathbf{X} \), where \( A = \begin{pmatrix} 1 & 1 \ -2 & -1 \end{pmatrix} \) and \( \mathbf{X} = \begin{pmatrix} x \ y \end{pmatrix} \).

Step 2: Find the Eigenvalues and Eigenvectors of A

To find the general solution, we compute the eigenvalues of matrix \( A = \begin{pmatrix} 1 & 1 \ -2 & -1 \end{pmatrix} \). The characteristic equation is \( \det(A - \lambda I) = 0 \), which simplifies to \( \lambda^2 + 0\lambda + 1 = 0 \). The eigenvalues are \( \lambda_1 = i \) and \( \lambda_2 = -i \).

Step 3: Find the General Solution

For complex conjugate eigenvalues \( \lambda = \pm i \), the general solution is of the form \( \mathbf{X}(t) = c_1 e^{it} \mathbf{v}_1 + c_2 e^{-it} \mathbf{v}_2 \). Calculate the eigenvectors associated with \( \lambda_1 = i \) using \( (A - iI)\mathbf{v} = 0 \). This results in the eigenvector \( \mathbf{v}_1 = \begin{pmatrix} 1 \ -2-i \end{pmatrix} \).

Step 4: Real Form of the General Solution

Transform the solution using Euler's formula: \( e^{it} = \cos(t) + i\sin(t) \). The real form of the solution for the system is \( \mathbf{X}(t) = \begin{pmatrix} c_1 \cos(t) - 2c_1 \sin(t) + c_2 \sin(t) \ c_1 \sin(t) + c_2 \cos(t) - 2c_2 \sin(t) \end{pmatrix} \).

Step 5: Periodicity Analysis

Solutions \( \cos(t) \) and \( \sin(t) \) are periodic with period \( 2\pi \). Thus, the general solution is periodic with period \( 2\pi \).

Step 6: Apply Initial Condition

Use the initial condition \( \mathbf{X}(0) = (-2,2) \). This gives the equations \( c_1(1) + c_2(0) = -2 \) and \( c_1(0) + c_2(1) = 2 \), which solve to \( c_1 = -2 \), \( c_2 = 2 \).

Step 7: Solution with Initial Condition

The particular solution with initial condition is \( \mathbf{X}(t) = \begin{pmatrix} -2 \cos(t) + 4\sin(t) \ -2\sin(t) + 2\cos(t) \end{pmatrix} \).

Step 8: Plotting the Solution

Using a graphing utility, plot \( x(t) = -2 \cos(t) + 4\sin(t) \) and \( y(t) = -2\sin(t) + 2\cos(t) \). Indicate the direction of time \( t \) increasing by marking points and arrows; the curve is an ellipse centered at the origin.

Key Concepts

Eigenvalues and Eigenvectors

In the context of a linear dynamical system, identifying eigenvalues and eigenvectors is crucial. They provide insights into the behavior of the system over time. Eigenvalues are a type of scalar, and they signify the factor by which the action of a particular transformation scales vectors in space. These vectors, known as eigenvectors, remain in the same line of action after transformation.
\( A = \begin{pmatrix} 1 & 1 \ -2 & -1 \end{pmatrix} \) is the matrix in our exercise, and solving the characteristic equation \( \det(A - \lambda I) = 0 \) was vital to finding its eigenvalues. This equation provided \( \lambda_1 = i \) and \( \lambda_2 = -i \). These values tell us how the system behaves, particularly that it involves oscillatory motion because they are purely imaginary.
Eigenvectors, which we calculated using \((A - \lambda I)\mathbf{v} = 0\), provide direction to the motion described by these eigenvalues. Understanding this allows us to analyze solutions to linear systems effectively.

Periodic Solutions

When dealing with linear dynamical systems, periodic solutions reveal patterns that repeat at regular intervals. In our given system, the periodicity is explored through trigonometric functions such as \( \cos(t) \) and \( \sin(t) \), which naturally cycle every \( 2\pi \).
Identifying trigonometric solutions in the real form helps confirm that the system repeats its trajectories over consistent periods. This property confirms that the general solution of our system is periodic with period \( 2\pi \).
This periodic behavior is crucial in understanding long-term predictions of the system's state without further complex calculations. Recognizing these patterns guides us in visualizing movements within the system, such as oscillations and circular motions.

Initial Conditions

Initial conditions in any differential equation help anchor the general solution to a specific context or state of the system at the beginning. For the problem at hand, the initial condition given was \( \mathbf{X}(0) = (-2, 2) \). Using these values allows us to solve for the constants that are part of the general solution.
These constants are crucial as they personalize the family of periodic solutions to one that fits the initial state. Here, it led us to find \( c_1 = -2 \) and \( c_2 = 2 \).
Applying initial conditions ensures that the particular solution aligns with the real-world scenario or data point we are working with, making theoretical results applicable and useful.

Matrix Differential Equations

Matrix differential equations simplify complex systems of equations by using matrices to neatly encapsulate and manipulate multiple linear equations simultaneously. Given the system \( x' = x + y \) and \( y' = -2x - y \), we used the matrix:
\[A = \begin{pmatrix} 1 & 1 \ -2 & -1 \end{pmatrix}\]
To express this system concisely as \( \mathbf{X}' = A\mathbf{X} \).
Solving these systems involves understanding matrix properties such as determinants and eigenvalues, which we have discussed earlier. Matrix forms bring organization and efficiency, making it easier to perform calculus operations, analyze stability, and solve for specific solutions.
This method is pivotal for engineers and scientists modeling real-world systems where multiple interrelated factors affect outcomes. Understanding matrix differential equations enriches students' analytical toolkit for tackling complex dynamics.