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IB Mathematics: Applications and Interpretation HL · AHL 9: Differential Equations · AHL 9.1

Name Date

Setting up and integrating Problem set B

Define the variables first. For a decrease put the minus sign in front of k. After integrating, always keep the constant c.

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. Write a differential equation for each situation. Define your variables first.

    1. (a) The number of subscribers to a channel grows at a rate proportional to that number.
    2. (b) The area of a spreading stain increases at a rate proportional to the fourth root of its area.

  2. Write a differential equation for each situation. Define your variables first.

    1. (a) The depth of silt in a harbour increases at a rate inversely proportional to the depth.
    2. (b) The length of a crack in a window increases at a rate inversely proportional to the square of its length.

  3. Write a differential equation for each situation. Define your variables first.

    1. (a) The volume of fuel in a lamp decreases at a rate proportional to the cube root of the volume.
    2. (b) The current in a coil decreases at a rate inversely proportional to the current.

  4. Find the general solution of each differential equation.

    1. (a) \(\dfrac{dy}{dx}=4x^{3} -\frac{10}{\sqrt{x}}\)
    2. (b) \(\dfrac{dy}{dt}=6e^{-\frac{t}{3}}\)
    3. (c) \(\dfrac{dy}{dt}=-3 + \frac{12}{t}\), \(t>0\)
    4. (d) \(\dfrac{ds}{dt}=3 \sin(4t)\)

In context

  1. A gym has 800 members. After \(t\) days, \(J\) of them have joined a step challenge. The rate at which \(J\) increases is proportional to the number who have joined and to the number who have not. Write a differential equation.

  2. A joke goes round a school of 900 students. After \(t\) minutes, \(J\) of them have heard it. The rate at which \(J\) increases is proportional to the number who have heard it and to the number who have not, and is inversely proportional to \(t\). Write a differential equation.

  3. A drone climbs with velocity \(v=\dfrac{dh}{dt}=5e^{-0.25t}\) m s⁻¹, where \(h\) is its height above the ground in metres and \(t\) is the time in seconds. It starts on the ground.

    1. (a) Find the general solution for \(h\) in terms of \(t\).
    2. (b) Given that \(h=0\) when \(t=0\), find the height after 8 seconds.

Stretch

  1. Rearrange each equation to make the derivative the subject, then find the general solution.

    1. (a) \(\dfrac{dF}{dm}+\frac{3}{m^{2}}=4\), \(m>0\)
    2. (b) \(\dfrac{dG}{dt}+6e^{-3t}=2\)

jmaths.xyz · AI HL · AHL 9.1 Setting up differential equations and integrating dy/dx = f(x)