Question

A complex solution of the differential equation \(\mathbf{y}^{\prime}=A \mathbf{y}\) is given, where \(A\) is a real \((2 \times 2)\) matrix. Let \(\mathbf{y}(t)\) denote any solution of \(\mathbf{y}^{\prime}=A \mathbf{y}\), where \(\mathbf{y}(0) \neq \mathbf{0}\). As \(t\) increases, how will the phase plane trajectory of the solution behave? Will the solution point (a) move around the origin on a circular orbit, (b) move around the origin on an elliptical orbit, (c) spiral inward toward the origin, or (d) spiral outward away from the origin?V

Short answer

(a) Circular orbit (b) Elliptical orbit (c) Inward spiral (d) Outward spiral Answer: To determine the qualitative behavior of the phase plane trajectories, find the eigenvalues of the given matrix \(A\) and verify their properties according to the conditions explained in the solution.

Step-by-step solution

Analyze the eigenvalues of the matrix A

Recall that the eigenvalues of a \((2\times2)\) matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \( are the roots of the characteristic equation \)\text{det}(A - \lambda I) = 0\(, where \)I\( is the identity matrix. The characteristic equation, in this case, is given by \)(a-\lambda)(d-\lambda)-bc$, which is a quadratic equation. Let \(\lambda_1, \lambda_2\) be the eigenvalues of \(A\). Their values could be real or complex, and they determine the behavior of the phase plane trajectories.

Identify the behavior of the phase plane trajectories

Now, based on the eigenvalues, we will determine whether the phase plane trajectories exhibit the behavior (a), (b), (c), or (d). (a) If both eigenvalues are purely imaginary and distinct (\(\lambda_1 = i\alpha, \lambda_2 = i\beta\), with \(\alpha\neq\beta\)), the phase plane trajectories will move around the origin on a circular orbit. (b) If both eigenvalues are imaginary with the same imaginary parts and a real part (\(\lambda_1 = \alpha + i\beta, \lambda_2 = \alpha - i\beta, \alpha\neq0\)), the phase plane trajectories will move around the origin on an elliptical orbit. (c) If both eigenvalues are complex with negative real parts (\(\text{Re}(\lambda_1) < 0, \text{Re}(\lambda_2) < 0\)), the phase plane trajectories will spiral inward toward the origin. (d) If both eigenvalues are complex with positive real parts (\(\text{Re}(\lambda_1) > 0, \text{Re}(\lambda_2) > 0\)), the phase plane trajectories will spiral outward away from the origin. To find out which case applies, one must analyze the given matrix \(A\) and find its eigenvalues. Then, based on their properties, the behavior of the phase plane trajectories can be determined.

Key concepts

Eigenvalues of Matrices

The concept of eigenvalues is fundamental in understanding the behavior of systems described by differential equations.

An eigenvalue of a matrix is a scalar that satisfies the equation \( Av = \lambda v \) where \( A \) is a matrix, \( v \) is a non-zero vector known as an eigenvector, and \( \lambda \) is the eigenvalue. In the context of differential equations, particularly those of the form \( \mathbf{y}' = A\mathbf{y} \) the eigenvalues of matrix \( A \) provide insights into how the solutions, or trajectories, will behave over time.

For a (2x2) matrix, the eigenvalues are found by solving the characteristic equation, which involves the determinant of \( A - \lambda I \) where \( I \) is the identity matrix. These values can be real, purely imaginary, or complex, and dictate how a solution will evolve. For example, purely imaginary eigenvalues signify oscillatory motion, whereas real eigenvalues may indicate growth or decay. Understanding eigenvalues is crucial as they serve as a bridge between linear algebra and differential equations, enabling us to predict the long-term behavior of dynamic systems.

Characteristic Equation

The characteristic equation is key to finding the eigenvalues of a matrix.

It is a polynomial equation derived from the matrix equation \( det(A - \lambda I) = 0 \), where \( A \) is the matrix in consideration, \( \lambda \) represents the eigenvalue, and \( I \) denotes the identity matrix. For a (2x2) matrix with elements \( a, b, c, \) and \( d \), the characteristic equation takes the form \( (a-\lambda)(d-\lambda) - bc = 0 \).

Solving this quadratic equation yields the eigenvalues, which are instrumental in analyzing the system's dynamics. Whether the solutions are real or complex, positive or negative, illustrates distinct trajectories and behaviors of the system. In systems modeled by differential equations, such as \( \mathbf{y}' = A\mathbf{y} \) understanding the characteristic equation enables us to predict future states and stability of the system.

Complex Solutions

Complex solutions arise in differential equations when the characteristic equation has complex roots.

These complex eigenvalues frequently occur in pairs of the form \( \lambda_1 = \alpha + i\beta \) and \( \lambda_2 = \alpha - i\beta \) where \( \alpha \) is the real part and \( \beta \) is the imaginary part, and \( i \) represents the square root of -1. The real part influences the rate of exponential growth or decay, while the imaginary part relates to the oscillatory motion.

When solving differential equations, complex eigenvalues indicate that the trajectories will exhibit spiral-like behavior. The direction (inward or outward) and shape (circular or elliptical) of these spirals are contingent on the sign and magnitude of the real and imaginary parts of the eigenvalues, respectively. Complex solutions embody the interplay between linear algebra and calculus, providing a more profound comprehension of dynamic systems.

Trajectory Behavior

Trajectory behavior in differential equations describes how solutions evolve over time in the phase plane.

The eigenvalues obtained from the characteristic equation of matrix \( A \) are the deciding factors for the trajectory behavior of a system \( \mathbf{y}' = A\mathbf{y} \). Different types of trajectories include circular orbits, elliptical orbits, and inward or outward spirals.

Circular Orbits

Purely imaginary eigenvalues, without real parts, lead to solutions that revolve around the origin, forming circular orbits. They represent systems with conservative oscillations, like undamped spring-mass systems.

Elliptical Orbits

Complex eigenvalues with identical imaginary parts and non-zero real parts create elliptical trajectories around the origin, often seen in damped oscillations.

Inward or Outward Spirals

Complex eigenvalues with negative real parts result in trajectories spiraling inward towards the origin, implying a stable system. Conversely, positive real parts suggest instability, as trajectories spiral outwards. These trajectory behaviors provide rich information on the nature and future behavior of dynamic systems.