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IB Mathematics: Applications and Interpretation HL · AHL 9: Differential Equations · AHL 9.10

Foci and centres Problem set A Answers

Skills

    1. (a) \(\lambda=1\pm 2i\)
    2. (b) Unstable focus (spiral out)
    1. (a) \(\dot x=6\); clockwise
    2. (b) Decreasing
    3. (c) Decreasing
    1. (a) Unstable focus (curves spiral out)
    2. (b) Stable focus (curves spiral in)
    3. (c) Neutral focus (closed ovals)

In context

    1. (a) \(\lambda=-1\pm 3i\)
    2. (b) Stable focus: the mass oscillates with smaller and smaller swings and comes to rest
    3. (c) \(\dot x=1>0\); clockwise
    1. (a) \(\lambda=0.5\pm1.94i\)
    2. (b) Decreasing
    3. (c) Unstable focus: the swings grow without bound, so the model cannot hold in the long term

Stretch

    1. (a) \(k=3\)
    2. (b) \(\lambda=\pm i\)
    3. (c) Anticlockwise
    1. (a) \(\lambda=-k\pm2i\)
    2. (b) k = 1.5: stable focus; k = 0: neutral focus; k = −1.5: unstable focus
    3. (c) Clockwise

jmaths.xyz · AI HL · AHL 9.10 Complex eigenvalues: foci and centres