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IB Mathematics: Applications and Interpretation HL · AHL 9: Differential Equations · AHL 9.11

Equilibrium points that are not the origin Problem set B Answers

Skills

    1. (a) (4, −1)
    2. (b) (1, 2)
    1. (a) (−1, 2)
    2. (b) \(\lambda=−1\pm3i\)
    3. (c) stable focus: the trajectories spiral towards \((−1, 2)\), so \(x\to−1\) and \(y\to2\)
    1. (a) Both rates are 0 at this point
    2. (b) (0, 0); (8, 0); (0, 4.5)

In context

    1. (a) (0, 0) and (250, 80)
    2. (b) (0, 0): both die out. (250, 80): both stay constant, with 2500 aphids and 800 ladybirds
    3. (c) \(x_1=260\), \(y_1=80.16\). Not an equilibrium: \(y\) changes
    1. (a) (20, 50)
    2. (b) \(\lambda=−0.6\) and \(\lambda=−0.3\): a stable node
    3. (c) 20 kg in the first and 50 kg in the second

Stretch

    1. (a) (2, −1)
    2. (b) \(\lambda=−2\): eigenvector \(\begin{pmatrix}1\\2\end{pmatrix}\), gradient 2; \(\lambda=−5\): eigenvector \(\begin{pmatrix}1\\−1\end{pmatrix}\), gradient −1
    3. (c) A stable node at (2, −1). Draw two straight lines through the point, of gradients 2 and −1. Arrows point towards the point on both lines. The curves meet the point tangent to the line for \(\lambda=−2\) and, far from it, run alongside the line for \(\lambda=−5\).
    1. (a) (2, 4)
    2. (b) \(k\gt6\)
    3. (c) The two conditions cannot both be true

jmaths.xyz · AI HL · AHL 9.11 Equilibrium points that are not the origin