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

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -9y - 14x;\quad \begin{pmatrix} 0 & 1 \\ -14 & -9 \end{pmatrix}\)
2. (b) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = 2y - 10x;\quad \begin{pmatrix} 0 & 1 \\ -10 & 2 \end{pmatrix}\)
3. (c) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = 7x - 3y;\quad \begin{pmatrix} 0 & 1 \\ 7 & -3 \end{pmatrix}\)

1. (a) stable: λ = −1 and −7
2. (b) unstable: λ = 2 and −5
3. (c) unstable: λ = 4 and 3
4. (d) unstable: λ = 7 and −7

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

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -3y - 36x;\quad \begin{pmatrix} 0 & 1 \\ -36 & -3 \end{pmatrix}\)
2. (b) stable: λ = −1.5 ± 5.81i
3. (c) stable: λ = −3 and −12

1. (a) λ = 0.3 ± 2.22i
2. (b) Unstable: the real part 0.3 is positive. This means that the temperature swings above and below the target by larger and larger amounts, so the system does not settle

1. (a) stable: λ = −1 ± 2.83i
2. (b) stable: λ = −1 and −9
3. (c) k = 0: not stable (real part 0). k negative: unstable (positive real part, or a positive real eigenvalue)

b > 0