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IB Mathematics: Applications and Interpretation HL · AHL 9: Differential Equations · AHL 9.17

Differential Equations: exam review Exam review Answers

Section A: short questions

    1. (a) \(\frac{dN}{dt}=kN(800-N)\)
    2. (b) \(y=3 x^{2} - 4 + \frac{4}{x}\)
    1. (a) \(e^{y}=\frac{x^{2}}{2}+c\)
    2. (b) \(y=\ln\left(\frac{x^{2}}{2}+e\right)\)
    3. (c) \(y=1.98\)
    1. (a) \(k=\frac{3}{25}=0.120\)
    2. (b) shown
    3. (c) \(5.78\text{ hours}\)
    4. (d) \(21.0\text{ hours}\)
    1. (a) \(T=20+70e^{-kt}\), \(k=0.0673\)
    2. (b) \(18.6\text{ minutes}\)
    1. (a) shown
    2. (b) \(m=100-90e^{-t/50}\)
    3. (c) \(40.5\text{ minutes}\)
    4. (d) \(100\text{ kg}\)
    1. (a) \(1\) and \(2\)
    2. (b) shown
    3. (c) a curve with a maximum at (0, 1), falling on both sides and staying below \(y=2\)
    4. (d) \(y\approx -0.72\) (accept a value within 0.3 of -0.72)
    1. (a) \(P=0\) and \(P=50\)
    2. (b) \(P=50\) is stable
    3. (c) \(P=25\)
    4. (d) \(5\)
    1. (a) \(\lambda=-1\pm \sqrt{6} i\ (\approx -1\pm 2.45i)\)
    2. (b) stable focus; spiral in towards the origin, so \(x\to0\) and \(y\to0\)
    3. (c) \(\frac{dx}{dt}=-5\): anticlockwise
    1. (a) \(\theta(0.2)\approx 0.5718\)
    2. (b) \(\theta(0.2)\approx 0.5700\)

Section B: long questions

    1. (a) \(y(1)\approx 1.25\)
    2. (b) \(y(1)\approx 1.48\)
    3. (c) \(y=\frac{1}{1 - \frac{x^{2}}{2}}\)
    4. (d) \(26.2\%\)
    5. (e) the curve is concave up; use a smaller step length
    1. (a) \(x_{n+1}=x_n+h(0.6x_n-0.01x_ny_n),\ y_{n+1}=y_n+h(0.004x_ny_n-0.3y_n)\)
    2. (b) \(x(1)\approx 120,\ y(1)\approx 44.9\)
    3. (c) \(x(4)\approx 109,\ y(4)\approx 84.2\)
    4. (d) \(92.2\) (at \(t=5.3\))
    5. (e) the Euler peaks do grow, but that is Euler error: the exact solution is a repeating cycle with peaks that do not grow, so the statement is not supported; a smaller step length reduces the growth
    1. (a) \(A=\begin{pmatrix}3&2\\1&4\end{pmatrix}\)
    2. (b) \(\lambda=2\) and \(\lambda=5\)
    3. (c) \(\lambda=2:\ \begin{pmatrix}2\\-1\end{pmatrix}\); \(\lambda=5:\ \begin{pmatrix}1\\1\end{pmatrix}\) (or multiples)
    4. (d) \(\begin{pmatrix}x\\y\end{pmatrix}=C\begin{pmatrix}2\\-1\end{pmatrix}e^{2 t}+D\begin{pmatrix}1\\1\end{pmatrix}e^{5 t}\)
    5. (e) \(x=20 e^{5 t} + 20 e^{2 t}\); \(y=20 e^{5 t} - 10 e^{2 t}\)
    6. (f) unstable node (both eigenvalues positive); see mark scheme for the sketch
    7. (g) \(y:x=1:1\)
    1. (a) \(\frac{dx}{dt}=y,\ \frac{dy}{dt}=- 3 x - 4 y;\ \begin{pmatrix}0&1\\-3&-4\end{pmatrix}\)
    2. (b) \(x(1)\approx 0.500\)
    3. (c) \(\lambda=-3\text{ or }-1\); stable
    4. (d) \(x=\frac{5 e^{- t}}{2} - \frac{e^{- 3 t}}{2}\)
    5. (e) \(44.1\%\)
    6. (f) (i) not stable; (ii) stable
    1. (a) \((2,3)\)
    2. (b) \(\lambda=-3.618\) or \(-1.382\); stable
    3. (c) \((2.10,3.80)\ne(2,4)\), so the claim is false
    4. (d) \(x\to2,\ y\to3\)

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