2025-11-21 Quiz 7
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1. Using Euler's method to approximate an ODE, you obtain the difference equation :math:`y_{n + 1} = y_n + h a y_n`.
What is the ODE?

  a) :math:`\frac{dy}{dt} = a y`

  b) :math:`\frac{dy}{dt} = a`

  c) :math:`\frac{dy}{dt} = h a y`

  d) :math:`\frac{dy}{dt} = h a`

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    <details><summary>Answer</summary>a</details>

2. A mass-spring problem is described by the ODE :math:`y'' + 2 y = cos \left( 2 t \right)`.
Which of the following is the equivalent first-order system?

  a) :math:`y' = x, \, y' = -2 y + cos \left( 2 t \right)`

  b) :math:`y' = x, \, x' + 2 y = cos \left( 2 t \right)`

  c) This second-order ODE can't be converted into a first-order system.

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    <details><summary>Answer</summary>b</details>

3. Draw a pair of graphs explaining the difference between forward Euler and backward (implicit) Euler for ODEs.
Do not forget to add labels to your charts.
Include a brief description explaining your graphs.
Hint: Where do you draw the tangent line?

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    <details><summary>Answer</summary>See the discussion on Wikipedia for <a href="https://en.wikipedia.org/wiki/Euler_method">forward Euler</a> and <a href="https://en.wikipedia.org/wiki/Backward_Euler_method">backward Euler</a>. Forward Euler is based upon a tangent line at the current location and backward Euler is based upon a tangent line at the next location.</details>