University of Notre Dame
Aerospace and Mechanical Engineering

AME 301: Differential Equations, Vibrations and Control
Homework 12

B. Goodwine
J. Lucey
Fall, 2003		 Issued: December 5, 2003
Due: December 10, 2003 (US)
11 December 2003 (UK)

  1. Consider

    \begin{displaymath} \xot x + x = cos(t) \quad x(0) = 1. \end{displaymath}

    1. Determine the exact solution (10 points).
    2. Using Euler's method, write a computer program in any language other than a matlab to determine an approximate solution to the equation (20 points).
    3. Using the fourth order Runge-Kutta method, write a computer program in any language other than matlab to determine an approximate solution to the equation (20 points).
    4. In addition to submitting your work for the exact solution and a listing of each program, submit the a figure plotting the exact solution, the solution from Euler's method and the solution from the fourth order Runge-Kutta method on the same plot from $t=0$ to $t=10$ (5 points each):
      1. $h = 1.0$
      2. $h = 0.5$
      3. $h = 0.2$
      4. $h = 0.1$

  2. Consider

    \begin{displaymath} \xot x + x = cos(t) + 5000 cos(100 \pi t) \quad x(0) = 1. \end{displaymath}

    1. Using Euler's method, write a computer program in any language other than a matlab to determine an approximate solution to the equation (5 points).
    2. Using the fourth order Runge-Kutta method, write a computer program in any language other than matlab to determine an approximate solution to the equation (5 points).
    3. In addition to submitting a listing of each program, submit a figure plotting the solution from Euler's method and the solution from the fourth order Runge-Kutta method on the same plot from t=0 to t=10 for (5 points each):
      1. $h = 1.0$
      2. $h = 0.1$
      3. $h = 0.01$
      4. $h = 0.001$
    4. For the same time steps (except for part (c)(i)), submit three plots where the time range is from $t=0$ to $t=0.5$ (10 points total).
    5. Explain what happens between $h = 0.1$ and $h = 0.01$ (20 points).


Last updated: December 5, 2003.
B. Goodwine (goodwine@controls.ame.nd.edu)
J. Lucey (jlucey@nd.edu)