Problem 3 - Advanced Engineering Mathematics

Question

The general solution of the linear system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}\) is given. (a) In each case discuss the nature of the solution in a neighborhood of \((0,0)\). (b) With the aid of a graphing utility plot the solution that satisfies \(\mathbf{X}(0)=(1,1)\) $$ \mathbf{A}=\left(\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right), \quad \mathbf{X}(t)=e^{t}\left[c_{1}\left(\begin{array}{c} -\sin t \\ \cos t \end{array}\right)+c_{2}\left(\begin{array}{c} \cos t \\ \sin t \end{array}\right)\right] $$

Step-by-Step Solution

Answer

The solution spirals outward from (0,0) as time increases, indicating growth and rotation.

Step 1: Determine Eigenvalues of Matrix A

We start by finding the eigenvalues of matrix \( \mathbf{A} \). The characteristic equation is given by \( \det(\mathbf{A} - \lambda\mathbf{I}) = 0 \). For \( \mathbf{A} = \begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix} \), the determinant becomes \( (1-\lambda)(1-\lambda) + 1 = \lambda^2 - 2\lambda + 2 = 0 \). Solving this quadratic equation, the eigenvalues are \( \lambda = 1 \pm i \).

Step 2: Interpret the Nature of the Solution

The eigenvalues \( \lambda = 1 \pm i \) indicate that the solution properties near the point \((0,0)\) show a spiral behavior due to the imaginary component and exponential growth due to the positive real part. Specifically, the solution will spiral outwards as \(t\) increases.

Step 3: Particular Solution with Given Initial Condition

Given \( \mathbf{X}(0) = (1,1) \), substitute \(t = 0\) into the general solution. We get \( \mathbf{X}(0) = c_1 \begin{pmatrix} 0 \ 1 \end{pmatrix} + c_2 \begin{pmatrix} 1 \ 0 \end{pmatrix} \). This simplifies to \( \begin{pmatrix} 1 \ 1 \end{pmatrix} = c_2 \begin{pmatrix} 1 \ 0 \end{pmatrix} + c_1 \begin{pmatrix} 0 \ 1 \end{pmatrix} \), leading to \( c_1 = 1 \) and \( c_2 = 1 \).

Step 4: Construct the Particular Solution

Substitute \( c_1 = 1 \) and \( c_2 = 1 \) back into the general solution to form the particular solution: \( \mathbf{X}(t) = e^t \left[ \begin{pmatrix} -\sin t \ \cos t \end{pmatrix} + \begin{pmatrix} \cos t \ \sin t \end{pmatrix} \right] = e^t \begin{pmatrix} \cos t - \sin t \ \sin t + \cos t \end{pmatrix} \).

Step 5: Plot the Solution

Using a graphing utility, plot the vector field and trajectory of the solution \( \mathbf{X}(t) = e^t \begin{pmatrix} \cos t - \sin t \ \sin t + \cos t \end{pmatrix} \) for \(t\) ranging over values that capture the spiral nature. Ensure the plot passes through the initial point \((1,1)\). This confirms the outward spiral behavior.

Key Concepts

Linear Systems

Linear systems of differential equations can be fundamental in understanding complex systems with multiple variables. A linear system involves equations where variables appear to the first power and are multiplied by constants. In simple terms, these systems can be visualized as groups of straight lines or planes intersecting in space.

In our case, the system given by \[ \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} \]is a two-dimensional linear system, meaning it deals with a vector \( \mathbf{X} \) that changes over time based on matrix \( \mathbf{A} \). This matrix \( \begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix} \) defines how different variables influence each other's rates of change. By studying these systems, you can predict how the state of the system evolves with time, which is essential in many fields such as engineering, physics, and economics.

Eigenvalues and Eigenvectors

In the study of linear systems, eigenvalues and eigenvectors are vital for understanding the system's behavior. They simplify complex calculations by transforming the system into a more understandable format.

Eigenvalues are the scalars that relate to how the system expands or contracts, while eigenvectors are the directions along which these changes happen. To find them, you solve the characteristic equation, \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \). In this problem, you find that our matrix \( \mathbf{A} \) leads to eigenvalues \( \lambda = 1 \pm i \).

• The real part (1) suggests exponential growth.
• The imaginary part (\( \pm i \)) leads to oscillatory behavior.

Put together, they indicate a spiral pattern. With these eigenvalues, we learn crucial details about the behavior of our system as time progresses.

Spiral Behavior

Spiral behavior in a linear system occurs when the system's movement traces a path that winds around a central point, either spiraling outward or inward. This behavior is often tied to the presence of imaginary parts in the eigenvalues, as we see in \( \lambda = 1 \pm i \).

In the context of our linear system, the real part 1 causes the solution to grow exponentially, meaning it expands outward from the point \((0,0)\). The imaginary part causes a continuous rotational motion, creating a spiral trajectory.

• A positive real part results in an outward spiral.
• A negative real part would lead to an inward spiral.

Understanding these properties helps explain how the system might behave over time, especially when plotting trajectories or analyzing stable states.

Trajectory Plotting

Plotting trajectories of a linear system can help visualize the evolution of solutions over time. In our scenario, trajectory plotting shows how the vector \( \mathbf{X}(t) \) changes, illustrating a path through space.

For the given initial condition \( \mathbf{X}(0) = (1,1) \), we substitute back successful values for \( c_1 \) and \( c_2 \), obtaining the particular solution. The resulting equation showcases the effect of combined sinusoidal and exponential terms, embodying the spiraling nature.

• Use graphing software to plot \( \mathbf{X}(t) = e^t \begin{pmatrix} \cos t - \sin t \ \sin t + \cos t \end{pmatrix} \).
• Scope over different \( t \) values to see how the trajectory evolves.

Such plots are pivotal for interpreting and predicting the multi-variable system's behavior, providing a clear graphical representation of theoretical solutions.