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IB Mathematics: Applications and Interpretation HL · AHL 9: Differential Equations · AHL 9.12

Euler's method for a coupled system Problem set A Answers

Skills

    1. (a) \(x_{n+1}=x_n+0.1(3x_n - y_n),\ y_{n+1}=y_n+0.1(x_n + 2y_n),\ t_n=0.1n\)
    2. (b) \(x_1=2.5,\ y_1=1.4\)
    3. (c) \(x_2=3.11,\ y_2=1.93\)
    1. (a) Oscillation with growing amplitude. First t-intercept at about \(t=0.61\); first minimum at about \((1.4,\ -5.65)\)
    2. (b) A spiral out from \((2,1)\), anticlockwise. It crosses the y-axis at about \(y=3.8\) and the negative x-axis at about \(x=-4.9\)

In context

    1. (a) \(x_{n+1}=x_n+0.1(-3y_n),\ y_{n+1}=y_n+0.1(2x_n - y_n),\ t_n=0.1n\)
    2. (b) First t-intercept at about \(t=0.55\); first minimum at about \((1.2,\ -2.4)\)
    3. (c) x oscillates with decreasing amplitude and tends to 0: the concentrations settle at their equilibrium values
    1. (a) A spiral out from \((2,0)\), clockwise. It crosses the y-axis at about \(y=-3.8\) and the negative x-axis at about \(x=-5.65\)
    2. (b) Both populations oscillate and the swings grow, so the model predicts larger and larger fluctuations

Stretch

    1. (a) Yes: x changes sign between 0 and 0.5, and again between 1 and 1.5
    2. (b) \(\lambda=-1\pm3i\) (p < 0: the values shrink, which agrees)
    3. (c) \(\pi\div q\approx1.05\). The two sign changes in the table are in intervals about 1.0 apart: it agrees
    1. (a) \(\lambda=\pm i\); closed curves round the origin (a neutral focus)
    2. (b) A spiral that slowly goes outwards: the distance from the origin grows from \(1\) to about \(2.70\)
    3. (c) Each Euler step moves along the tangent, which carries the point a little further out, and the error builds up

jmaths.xyz · AI HL · AHL 9.12 Euler's method for a coupled system: graphs and phase portraits