Final answer:
To determine the stability of the equilibrium for the system x' = Ax, the roots of the characteristic polynomial p_A(λ) are examined. Positive real parts indicate instability, while negative or non-positive real parts indicate stability or marginal stability, respectively. The given polynomials show a mix of stable, unstable, and marginally stable equilibria.
Step-by-step explanation:
To classify the stability of an equilibrium solution for the system x' = Ax given the characteristic polynomials p_A(λ), we need to examine the roots of each polynomial. The stability depends on the signs of the real parts of the eigenvalues (roots).
- Stable if all real parts are negative,
- Unstable if any real part is positive,
- Stable or marginal stability if real parts are non-positive but there may be complex eigenvalues with zero real part.
For (a) p_A(λ) = λ² − 8λ + 12, we find roots by factoring: (λ - 2)(λ - 6), which are λ1 = 2 and λ2 = 6, both positive, indicating unstable equilibrium.
For (b) p_A(λ) = λ² + 64 has roots λ1,2 = ±8i, purely imaginary, suggesting marginal stability since there are no real parts.
For (c) p_A(λ) = λ² − 5λ − 14 factors into (λ - 7)(λ + 2), yielding roots λ1 = 7 and λ2 = -2, with a positive real part indicating unstable equilibrium.
For (d) p_A(λ) = λ² − 6λ + 25 does not factor nicely, we use the quadratic formula and find complex roots with positive real parts, indicating unstable equilibrium.
For (e) p_A(λ) = λ² + 8λ + 25 factors into (λ + 5)², yielding a repeated negative root λ = -5, suggesting stable equilibrium.
For (f) p_A(λ) = λ² + 3λ + 2 factors into (λ + 1)(λ + 2), with roots λ1 = -1 and λ2 = -2, both negative, indicating stable equilibrium.