The problem in the image appears to involve analyzing a linear system and determining the type and stability of its critical points. Here’s a breakdown of the tasks in question Q2:

1. Problem Statement:
   The system is given in matrix form: $\begin{pmatrix} x' \\ y' \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix}$ where $A = \begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix}$.

2. Tasks:

   • (a) Determine the type of critical point and its stability.
   • (b) Solve the system.

Solution Approach

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

To determine the type and stability of the critical point at the origin, we need to analyze the eigenvalues of the matrix $A$.

1. Find the Eigenvalues of $A$:

   • The characteristic polynomial of $A$ is given by $\det(A - \lambda I) = 0$.
   • Calculate $\det \begin{pmatrix} -\lambda & 1 \\ -1 & -1 - \lambda \end{pmatrix}$.
   • Solve for $\lambda$ to find the eigenvalues.

2. Determine the Type:

   • Based on the nature of the eigenvalues (real vs complex and positive vs negative real part), we can classify the critical point as a node, saddle, spiral, or center.

3. Determine Stability:

   • Stability is determined by whether all eigenvalues have negative real parts (indicating stability) or if there are any eigenvalues with positive real parts (indicating instability).

(b) Solve the System

1. Use Eigenvalues and Eigenvectors:
   • Using the eigenvalues and eigenvectors of $A$, form the general solution to the system.
   • Express the solution in terms of $x(t)$ and $y(t)$ using the eigenvector decomposition method.

Would you like a detailed step-by-step solution for each part?

Here are five related questions that can help deepen your understanding:

1. How are the eigenvalues of a matrix related to the stability of its critical points?
2. What does it mean for a critical point to be a spiral, node, or saddle?
3. Can a system have a stable and an unstable critical point simultaneously?
4. How does the trace and determinant of a matrix help in determining the type of equilibrium?
5. What is the significance of complex eigenvalues in linear systems?

Tip: When analyzing stability, always pay close attention to the real part of the eigenvalues, as it directly impacts whether a solution diverges or converges over time.