The problem presented in the image involves analyzing a system of differential equations to determine the type of critical point and its stability. Here’s a breakdown of the problem and solution approach:

Given System

The system of differential equations is represented in matrix form as:

$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$

This system can be written as: $x' = y$ $y' = -x$

Part (a): Determine the Type of Critical Point and Stability

1. Find the Eigenvalues: To classify the critical point, we need to find the eigenvalues of the matrix $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.

   The eigenvalues $\lambda$ satisfy the characteristic equation: $\det(A - \lambda I) = 0$

   Calculating the determinant for $A - \lambda I$: $\det \begin{pmatrix} -\lambda & 1 \\ -1 & -\lambda \end{pmatrix} = \lambda^2 + 1 = 0$

   Solving $\lambda^2 = -1$ gives $\lambda = \pm i$ (purely imaginary eigenvalues).

2. Interpretation of Eigenvalues:

   • Purely imaginary eigenvalues $\pm i$ indicate a center type critical point.
   • Since there are no real parts to the eigenvalues, this critical point is stable (but not asymptotically stable), meaning trajectories will neither converge to nor diverge from the critical point but will form closed orbits around it.

Part (b): Solve the System

To solve the system, we can use the eigenvalues and eigenvectors approach or recognize it as a simple harmonic oscillator. The general solution will involve trigonometric functions due to the imaginary eigenvalues.

Would you like a detailed solution for part (b) or have any questions about part (a)?

Related Questions

1. How are centers and saddle points different in stability analysis?
2. What does it mean for a critical point to be asymptotically stable?
3. How do eigenvalues determine the stability of a system?
4. Can a system with complex eigenvalues be unstable? Under what conditions?
5. What is the physical interpretation of a system with imaginary eigenvalues?

Tip

When analyzing linear systems, finding the eigenvalues is a crucial first step in determining the nature and stability of equilibrium points.