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IB Mathematics: Applications and Interpretation HL · AHL 9: Differential Equations · AHL 9.5

Name Date

Euler's method Problem set B

Write the formulas, then use your calculator to run the steps. Give the final answer to 3 significant figures.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=2y\) and \(y(0)=1.5\).

  2. Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=y-3x\) and \(y(0)=2\).

  3. Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=2x+y\) and \(y(0)=−1\).

  4. Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=\dfrac{2x}{y}\) and \(y(0)=3\).

  5. Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=x^2y\) and \(y(0)=2\).

  6. Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=ye^{-2x}\) and \(y(0)=2\).

In context

  1. The mass \(m\) grams of a radioactive sample satisfies \(\dfrac{dm}{dt}=-0.5m\) with \(m(0)=6\), where \(t\) is the time in days. Use Euler's method with \(h=0.2\) to estimate \(m(1)\).

  2. The number \(R\) (in hundreds) of students who know a rumour satisfies \(\dfrac{dR}{dt}=0.4R\left(1-\dfrac{R}{5}\right)\) with \(R(0)=0.5\), where \(t\) is the time in hours. Use Euler's method with \(h=0.2\) to estimate \(R(1)\).

Stretch

  1. A differential equation is \(\dfrac{dy}{dx}=x-ky\), where \(k\) is a constant, and \(y(0)=5\). Euler's method with \(h=0.2\) gives \(y_1=4.2\).

    1. (a) Find the value of \(k\).
    2. (b) Find \(y_2\).

jmaths.xyz · AI HL · AHL 9.5 Euler's method for dy/dx = f(x, y)