Mixed practice: stability of second order equations Problem set B Answers

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -11y - 18x;\quad \begin{pmatrix} 0 & 1 \\ -18 & -11 \end{pmatrix}\)
2. (b) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = 6y - 25x;\quad \begin{pmatrix} 0 & 1 \\ -25 & 6 \end{pmatrix}\)
3. (c) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = 8x - 5y;\quad \begin{pmatrix} 0 & 1 \\ 8 & -5 \end{pmatrix}\)

1. (a) stable: λ = −1 and −9
2. (b) unstable: λ = 3 and −5
3. (c) unstable: λ = 5 and 3
4. (d) unstable: λ = 8 and −8

1. (a) stable: λ = −3 ± 5i
2. (b) unstable: λ = 2 ± 4i
3. (c) not stable: λ = ±5i

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -4y - 49x;\quad \begin{pmatrix} 0 & 1 \\ -49 & -4 \end{pmatrix}\)
2. (b) stable: λ = −2 ± 6.71i
3. (c) stable: λ = −2.86 and −17.1

1. (a) λ = 0.4 ± 2.62i
2. (b) Unstable: the real part 0.4 is positive. This means that the population swings above and below the target by larger and larger amounts, so it does not settle

1. (a) stable: λ = −1 ± 3.87i
2. (b) stable: λ = −2 and −8
3. (c) k = 0: not stable (real part 0). k negative: unstable (positive real part, or a positive real eigenvalue)

a > 0