2025-11-21 Quiz 7

1. Using Euler’s method to approximate an ODE, you obtain the difference equation \(y_{n + 1} = y_n + h a y_n\). What is the ODE?

  1. \(\frac{dy}{dt} = a y\)

  2. \(\frac{dy}{dt} = a\)

  3. \(\frac{dy}{dt} = h a y\)

  4. \(\frac{dy}{dt} = h a\)

Answera

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

  1. \(y' = x, \, y' = -2 y + cos \left( 2 t \right)\)

  2. \(y' = x, \, x' + 2 y = cos \left( 2 t \right)\)

  3. This second-order ODE can’t be converted into a first-order system.

Answerb

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?

AnswerSee the discussion on Wikipedia for forward Euler and backward Euler. 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.