Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-hl/differential-equations/ahl-9-11/problems/

jmaths

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

Equilibrium points that are not the origin Problem set A Answers

Skills

    1. (a) (3, 1)
    2. (b) (−2, −1)
    1. (a) (4, 3)
    2. (b) \(\lambda=−3\) and \(\lambda=−1\)
    3. (c) stable node: the trajectories approach \((4, 3)\), so \(x\to4\) and \(y\to3\)
    1. (a) Both rates are 0 at this point
    2. (b) (0, 0); (10, 0); (0, 12)

In context

    1. (a) (0, 0) and (75, 25)
    2. (b) (0, 0): both die out. (75, 25): both stay constant, with 7500 hares and 2500 foxes
    3. (c) \(x_1=80\), \(y_1=25.05\). Not an equilibrium: \(y\) changes
    1. (a) (30, 40)
    2. (b) \(\lambda=−0.8\) and \(\lambda=−0.3\): a stable node
    3. (c) 30 kg in the first and 40 kg in the second

Stretch

    1. (a) (1, 2)
    2. (b) \(\lambda=1\): eigenvector \(\begin{pmatrix}1\\−2\end{pmatrix}\), gradient −2; \(\lambda=4\): eigenvector \(\begin{pmatrix}1\\1\end{pmatrix}\), gradient 1
    3. (c) An unstable node at (1, 2). Draw two straight lines through the point, of gradients −2 and 1. Arrows point away from the point on both lines. The curves meet the point tangent to the line for \(\lambda=1\) and, far from it, run alongside the line for \(\lambda=4\).
    1. (a) (1, 5)
    2. (b) \(k\gt2\)
    3. (c) The two conditions cannot both be true

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